# Statistical estimation using confidence intervals

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 0894PP_ch06 15/3/02 11:02 am Page Statistical estimation using confidence intervals In Chapter 2, the concept of the central nature and variability of data and the methods by which these two phenomena may be mathematically determined were described. As previously explained, one of the main purposes for calculating the central nature (e.g. the mean) and the variability (e.g. the standard deviation) of a set of sample data is to gain an understanding of the corresponding population statistics. In other words, the mean ( KX ) and standard deviation (s) of sample data are employed to estimate the true (population) mean and true (population) standard deviation. However, it is reasonable to ask, how reliable are the sample data at representing the population data? Is the sample data a good estimation of the population data? 6.1 The concept of confidence intervals After calculating the mean and standard deviation of a sample, as is the normal approach in the pharmaceutical and related sciences, we need to provide an indication of the reliability of the data. For example, in a clinical trial (N # 20 patients) the volume of urine produced after the administration of a new diuretic drug has been calculated as 5.2 & 1.9 L. In light of the small sample size, it is unreasonable to predict that the population mean and standard deviation will be identical to these observed sample values, as each sample will produce different mean and standard deviation values. Therefore, when reporting the mean of sample data it is good practice to present some indication of the reliability of the data, i.e. the quality of the estimation of the true mean from the sample mean. This is performed using confidence intervals. The confidence intervals are quoted as a mean and range (interval), the latter representing the probability of observing the true mean. The imposed probability is chosen by the person responsible for the statistical manipulation of data; however, the most frequently employed confidence intervals are 90%, 95% and 99%. Confidence intervals are demonstrated graphically in Figures 6.1 and 6.2. In these, 10 samples, each containing 10 observations, were

2 0894PP_ch06 15/3/02 11:02 am Page Statistical estimation using confidence intervals selected from a normal distribution, whose mean was known to be 50. For each of the 10 samples, the mean and 90% confidence intervals were calculated. (The method for doing this is shown in the following section and does not need to be explained at this stage.) An example of the outcome for each sample is shown in Figure 6.1. The central portion of the bar denotes the mean that has been calculated using the sample data. The regions that extend equally beyond the mean in either direction are the calculated 90% confidence intervals. As for any range, there are upper and lower limits. The calculated mean and (90%) confidence intervals for each sample may be plotted on the one graph, as shown in Figure 6.2. The mean of the population has been included in this graph as a dashed line. Some features of Figure 6.2 require clarification to enable the reader to fully understand the basis of confidence intervals. First, it may be observed that, as expected, the means of all the different samples, although numerically close, are not equal to the population mean. Furthermore, the means of the individual samples are not numerically equal, once more reflecting sample variability. Secondly, the confidence intervals of the individual groups also differ in magnitude. These differences may also be logically explained. Confidence intervals are calculated using the standard deviation of the sample and therefore, if the variability of a sample is large, then the size of the confidence interval (i.e. the range) will be larger. Consequently, differences in the sizes of the confidence intervals reflect differences in the standard deviations of the individual samples. Finally, it may be observed from Figure 6.2 that, although the individual sample means do not exactly equal the population mean, the 90% confidence intervals in all but one case (denoted by an asterisk) do encompass the population mean. Therefore, the calculated 90% confidence intervals have included the population mean in 9 out of 10 observations, i.e. 90% of observations. If 90% confidence interval range for the individual sample Lower Upper Mean of sample (N = 10) Figure 6.1 Mean and confidence intervals.

3 0894PP_ch06 15/3/02 11:02 am Page 137 Confidence intervals for the population mean * Arbitrary units = Mean ± confidence intervals Figure 6.2 Mean and 90% confidence intervals derived from 10 samples (10 observations per sample group). the 95% confidence intervals had been calculated for each sample and plotted as described in Figure 6.2, the confidence intervals would have encompassed the true mean in 19 out of 20 samples (95%), and so on. Confidence intervals are therefore calculated to provide the user with the probability that a single sample will contain the true mean (or indeed true standard deviation, etc). It is still possible that if a sample is removed and the mean & 95% confidence intervals calculated, the true mean will not fall within the range calculated. However, it is also true that if another 19 samples were removed and the mean & 95% confidence intervals calculated, the true mean would fall within the confidence intervals described by these 19 samples. The calculation of confidence intervals is performed with reference to a probability distribution, e.g. the normal distribution, t distribution or χ 2 distribution. The mechanism for performing such calculations forms the basis of the remainder of this chapter. 6.2 Confidence intervals for the population mean and the normal distribution One of the most frequently used methods for calculating confidence intervals involves the use of the normal distribution. If the data conforms to the normal distribution, the two-tailed confidence interval may be calculated using the following equation: P% = KX & z P% σ N

4 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals where P% is the selected confidence interval, i.e. usually 90%, 95% or 99%, KX is the observed mean, σ is the population standard deviation and z P% is the z value corresponding to the percentage confidence interval. Remember that the term σ N refers to the standard error of the mean, previously introduced in Chapter 2. The z value is therefore an important parameter in the calculation of confidence intervals. Confidence limits are two-tailed events as there are two possible outcomes, i.e. one interval below and one interval above the mean. The z value chosen for inclusion in the equation or calculation of confidence is dependent on the selected value of the probability, as shown in Figure 6.3. Therefore, if we want to calculate the 95% confidence interval, we choose a z value of 1.96 as this value corresponds to the area under the standard normal distribution that encompasses 95% of all values. Similarly, 2.58 and 1.65 are chosen when 99% and 90% confidence intervals are to be calculated. The calculation of confidence intervals based on the standardised normal distribution is shown in the following example. EXAMPLE 6.1 A clinical trial (N # 30 patients) has been performed in which the volume of distribution of a new anti-diabetes drug has been calculated as 10.2 & 1.9 L. Calculate the 95% and 99% confidence limits of the mean value (assuming that the data originated from a normal distribution). 95% confidence interval. From the standardised normal distribution, the z statistic associated with 95% probability (two-tailed) is Therefore the 95% confidence interval is calculated as follows: P 95% = KX & z 95% σ 1.9 = 10.2 & 1.96 = 10.2 & 0.68 L N 30 = L 99% confidence interval. Similarly, the 99% confidence interval is calculated using a z value of 2.58 (corresponding to 99% probability in a two-tailed outcome): P 99% = KX & z 99% σ 1.9 = 10.2 & 2.58 = 10.2 & 0.89 L N 30 = L Having calculated these values, it is important at this point for the reader to fully comprehend the meaning of confidence intervals. In the clinical example described, the 95% confidence interval was 10.2 & 0.68 L, i.e L. The observed mean value (10.2 L) is unlikely to be identical to the population mean value. However, the

5 0894PP_ch06 15/3/02 11:03 am Page 139 Confidence intervals for the population mean 139 (a) (b) % % (c) % Figure 6.3 Probability functions within the standardised normal distribution. calculated confidence interval provides an estimation of the reliability of the measured mean. Therefore, we are 95% certain that the true mean will lie within the range defined by the confidence intervals, i.e L. In other words, if 100 samples were selected and their means and confidence intervals calculated, it is likely that 95 such confidence intervals would contain the true mean. Similarly, in a 99% confidence interval, there is a 99% chance that the true mean will lie within the defined (calculated) range. In general laboratory practice, the mean and confidence intervals of one set of data (sample) are normally calculated, the use of confidence intervals providing sufficient information about the reliability of the mean. Students often ask why the 99% confidence interval encompasses a greater range of values than the corresponding 95% interval. The answer is quite straightforward and concerns an appreciation of the role of probability in the calculation of confidence intervals. In the calculation of confidence intervals, reference is made to the appropriate probability distribution, the normal distribution in our case. As described previously, the total area under the normal distribution curve is 1 (equal to 100% probability). The area under the curve (and the z value) increases proportionally as the probability of an event increases. Hence the z value associated with 99% probability, i.e. 2.58, is greater than that for the 95% probability (1.96) and accordingly, to ensure 99%

6 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals probability of an event, a greater spread of z values is required. Thus in the calculation of 99% confidence intervals a greater area under the normal curve is being considered than with 95% confidence intervals and hence the range of values within the 99% confidence interval exceeds that of the lower interval. The calculation and applications of confidence intervals are further explored in the following examples. EXAMPLE 6.2 The aluminium content of L samples of intravenous fluids was analysed and the mean and standard deviation calculated as 50 & 3 ppm. Calculate the 95% and 99% confidence limits for estimating the true mean aluminium concentration. 95% confidence interval. The sample size is large (1000), so it may be assumed that the aluminium concentration conforms to normality and therefore, the following equation may be employed to calculate confidence intervals: P 95% = KX & z 95% σ N where Z 95% denotes the z value corresponding to a probability of 95% (twotailed), i.e. 1.96, σ # 3 ppm and N # Thus: P 95% = 50 & = 50 & 0.19 ppm 1000 The 95% confidence limits are ppm, i.e. there is a 95% chance that the true population mean will be encompassed within this range. 99% confidence interval. Once more, the following equation may be used: P 99% = KX & z 99% σ N where z 99% denotes the z value corresponding to a probability of 99% (twotailed), i.e. 2.58, σ # 3 ppm and N # Thus: P 99% = 50 & = 50 & The 99% confidence limits are ppm, i.e. there is a 99% chance that the true population mean lies within this region. EXAMPLE transdermal patches have been removed from a batch and the stability of the active agent determined at 25 C. The observed mean (and standard deviation) degradation rate constant for the therapeutic agent was 0.09 & 0.01 day 01. Calculate the 80% and 95% confidence intervals of the estimates of the mean, and the confidence that the mean degradation rate of the population is 0.09 & h 01.

7 0894PP_ch06 15/3/02 11:03 am Page 141 Confidence intervals for differences between means % confidence interval P 80% = KX & z 80% σ N From the standardised normal distribution, the z value associated with 80% probability is 1.28 (Appendix 1). Therefore P 80% = KX & z 80% σ N The 80% confidence interval for the estimation of the population mean is h % confidence interval P 95% = KX & z 95% σ N The 95% confidence interval for the estimation of the population mean is h 01. Confidence that the mean degradation rate of the population is 0.09 & h 01. This question offers a different perspective on the use of confidence intervals. In the previous examples we have defined the probability of the confidence interval and then calculated the interval from this. In this example, the interval has been defined and the confidence of that interval is unknown and requires calculation. Once more the generic equation is employed: P% = KX & z P% σ N 0.01 = 0.09 & 1.28 = 0.09 & h = 0.09 & 1.96 = 0.09 & h The known parameters are inserted into this equation: P% = 0.09 & z P% 0.01 = 0.09 & (z P% 0.001) & = 0.09 & 0.001z P% Therefore: = 0.001Z 1.5 = z P% The area under the standardised normal curve from z # 0 to z # 1.5 is Similarly the area under the standardised normal distribution from z # 0 to z #01.5 is Therefore, the degree of confidence is , i.e %. 6.3 Confidence intervals for differences between means Most frequently, confidence intervals are calculated to provide an estimation of the mean of a population, as described in section 6.2. However, in many cases, it is useful to describe the confidence intervals on the differences between means. Once more, the normal distribution

8 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals may be employed for this purpose, i.e. it is assumed that the means are derived from a normal distribution. The equation used here is based on the one employed for the calculation of the confidence intervals of a single mean, with two exceptions: the difference between the means replaces the mean value the standard error of differences between means is used in place of the standard error of the mean (i.e. the square root of the sum of the squares of their standard errors) The modified equation is: P% = (KX 1 KX 2 ) & z P% s s 2 2 N 1 N 2 where KX and are the means of samples 1 and 2, respectively; s 2 1 KX 2 and s are the variances of samples 1 and 2, respectively; z P% is the z value relating to the chosen level of probability (e.g. 90%, 95%, 99%), N 1 and N 2 are the sample sizes in samples 1 and 2, respectively and s s 2 2 N 1 N 2 is the standard error of the difference between two independent means. The use of this equation is illustrated in the following examples. EXAMPLE 6.4 A quality control laboratory wishes to examine and compare the mechanical properties of two wound dressings using tensile analysis. Accordingly, the mean (& standard deviation) tensile strength of a proprietary wound dressing was examined (N # 250) and recorded as & 0.57 MPa. The mean (& standard deviation) of a generic wound dressing was 8.99 & 0.73 MPa (N # 150). Calculate the 90% and 95% confidence intervals of the difference between the tensile strengths of the two wound dressings. 90% confidence interval. From the information given, we know that KX 1 and KX 2 are and 8.99 MPa, respectively; s 1 and s 2 are 0.57 and 0.73 MPa, respectively; z P% is the z value relating to the chosen level of probability, i.e for a 90% confidence interval; and N 1 and N 2 are 250 and 150, respectively. Inserting this information into the relevant equation enables us to calculate the 90% confidence interval: P 90% = (KX 1 KX 2 ) & z 90% s 2 1 N 1 + s 2 2 N 2 (0.57) 2 = ( ) & (0.73) = 1.36 & 0.12 MPa

9 0894PP_ch06 15/3/02 11:03 am Page 143 Confidence intervals for differences between means 143 Therefore, the 90% confidence interval for the difference between the mean tensile strength of the two wound dressings is MPa. Therefore, there is a 90% probability that the true mean difference in tensile strengths between the two dressings may be found in the range MPa. 95% confidence interval. The various parameters in the equation below are identical to those described above with the exception of the z value, which now reflects 95% probability, i.e. 1.96: P 95% = (KX 1 KX 2 ) & z 95% s 2 1 N 1 + s 2 2 N 2 (0.57) 2 = ( ) & (0.73) = 1.36 & 0.14 MPa Therefore, the 95% confidence interval for the difference between the mean tensile strength of the two wound dressings is MPa and there is a 95% probability that the true mean difference in tensile strengths between the two dressings may be found in this range. EXAMPLE 6.5 A new therapeutic agent has been developed to promote diuresis. In a clinical trial, the diuretic effects of this new agent and a commercial agent were assessed. In this the urine was collected over a 12-h period after administration of a single tablet. The mean (& standard deviation) volume of urine collected in the group of patients (N # 65) who received the new therapeutic agent was 48.8 & 9.1 L, whereas in the control group (who received the commercially available preparation) the volume was 37.9 & 4.6 L (N # 95). Calculate the 95% and 99% confidence intervals for the difference in urine volume induced by the two therapeutic agents. 95% confidence interval. Once more, the equation described at the start of section 6.3 is employed to calculate the required confidence interval, using z 95% # 1.96: P 95% = (KX 1 KX 2 ) & z 95% s 2 1 N 1 + s 2 2 N 2 = ( ) & 1.96 (9.1) 2 = 10.9 & 2.40 L 65 + (4.6) 2 95 Therefore, the 95% confidence interval for the difference in mean urine volume is L.

10 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals 99% confidence interval. Employing z 99% # 2.58, the above equation may be written as: P 99% = (KX 1 KX 2 ) & z 99% s 2 1 N 1 + s 2 2 N 2 = ( ) & 2.58 (9.1) 2 = 10.9 & 3.16 L 65 + (4.6) 2 95 The confidence intervals for the differences between means is interpreted in the same way as other confidence intervals. Therefore, in this example, there is a 99% probability that the true difference between the two diuretics will fall within the range L. 6.4 Confidence intervals for standard deviations Determination of the confidence intervals for the standard deviation is most frequently employed to examine the variability of data, e.g. whenever there is unexpectedly high variation in a sample. For this purpose, the χ 2 distribution is used to calculate confidence intervals for the population standard deviation. From Appendix 3, χ 2 values may be obtained that provide information concerning the areas under the probability curve. Hence, to calculate the 95% confidence interval of the population standard deviation, the region under the χ 2 distribution that equates to 95% is found between χ 2 and. Therefore, 2.5% of observations lie both below χ χ and above χ With this in mind, the 95% confidence interval is calculated using the following equation: χ 2 1)s ` (N ` χ 2 σ where N 0 1 is the number of degrees of freedom, i.e. the sample size minus 1, s is the sample standard deviation, σ is the population standard deviation, and χ and χ are the χ 2 statistics relating to probabilities of 2.5% and 97.5%, respectively, for n 0 1 degrees of freedom. This equation may be more conveniently rearranged to give s (N 1) s (N 1) ` σ ` χ χ An example of the use of the χ 2 distribution to calculate the confidence

11 0894PP_ch06 15/3/02 11:03 am Page 145 Confidence intervals for standard deviations 145 intervals associated with the population standard deviation is provided next. EXAMPLE 6.6 Before the start of a clinical trial, the weights of 20 healthy male volunteers (aged years) were measured and the mean and standard deviation recorded as 76.1 & 9.4 kg. Calculate the 90%, 95% and 99% confidence intervals of the population standard deviation. 90% confidence intervals. Using the equation described above, s (N 1) s (N 1) ` σ ` χ 0.95 χ 0.05 where N is the number of volunteers (20); s is the sample standard deviation, i.e. 9.4 kg; χ 0.95 for 19 degrees of freedom (i.e. 20 volunteers 0 1) is χ , i.e = 5.49; χ 0.05 for 19 degrees of freedom is χ , i.e = Therefore: s (N 1) s (N 1) ` σ ` = ` σ ` χ 0.95 χ ` σ ` kg Therefore, there is a 90% chance that the true population standard deviation (weights of healthy men aged years) will reside within the above limits. 95% confidence intervals. Using the appropriate equation: s (N 1) s (N 1) ` σ ` χ χ where N and s are as before; χ for 19 degrees of freedom is χ , i.e = 5.74; and χ for 19 degrees of freedom is χ , i.e = Therefore: s (N 1) s (N 1) ` σ ` = ` σ ` χ χ ` σ ` kg and there is a 95% chance that the true population standard deviation falls within the above limits. 99% confidence intervals. As before: where N and s is the sample standard deviation; χ for 19 degrees of freedom is, i.e = 6.21; and χ for 19 degrees of freedom is, i.e. χ s (N 1) s (N 1) ` σ ` χ χ χ

12 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals 6.84 = Therefore: s (N 1) s (N 1) ` σ ` = ` σ χ χ ` σ ` Kg and there is a 99% chance that the true population standard deviation (weights of healthy men aged years) will lie between 6.60 and kg. It is worth pointing out that the confidence limits of the true (population) standard deviation from the sample standard deviation may be estimated using the normal distribution. In this it is assumed that the population from which the sample data has been derived is normally distributed. When the sample size is large, i.e. greater than 30, the mathematical function 2χ 2 2(N 1) 1 = z p is appropriate and approximates to a normal distribution. Therefore, by rearrangement of the equation to produce χ 2 = 1, the χ 2 2 (z p + 2(N 1) 1) 2 statistic may be calculated using the normal distribution. From this, the confidence intervals of the true standard deviation may be calculated from the sample standard deviation using the χ 2 statistic, as described previously. Alternatively, the standard deviation may be directly calculated from the normal distribution using the following equation: s P% = s & z p 2N where s is the sample standard deviation, N is the number of samples and z P is the z value corresponding to the desired level of probability. The uses of the different methods for calculating the confidence intervals of the standard deviation are described in the next example. EXAMPLE 6.7 A solid dosage form containing 100 mg of a therapeutic agent has been manufactured and stored for 1 year at 37 C and 75% relative humidity. Analysis of the concentration of therapeutic agent in 101 tablets following this period of storage revealed that the mean (& standard deviation) drug content was & 4.56 mg. Calculate the 95% confidence interval of the population standard deviation of the concentration of therapeutic agent within the tablets. Use of the χ 2 distribution s (N) ` σ ` s (N) = ` σ ` χ χ ` σ ` 5.32 mg Therefore, the 95% confidence interval of the population standard deviation

13 0894PP_ch06 15/3/02 11:03 am Page 147 Confidence intervals for proportions 147 of drug content in solid dosage forms is mg (Note: N is used in place of N 0 1 as the sample size is large). Use of the normal distribution. As described above, χ 2 = 1 2 (z p + 2(N 1) 1) 2 Therefore: χ = 1 2 (z p + 2(N 1) 1) 2 = 1 ( (101) 1) 2 2 = χ = 1 2 (z p + 2(N 1) 1) 2 = 1 ( (101) 1) 2 2 = Note that the values of χ 2 calculated using the normal distribution are numerically similar to the values derived from the χ 2 distribution (Appendix 3). To calculate the confidence intervals as requested, these χ 2 values are used, as described previously: s (N) ` σ ` s (N) = ` σ ` χ χ ` σ ` 5.30 mg Results calculated using either method are similar. 6.5 Confidence intervals for proportions Confidence intervals of proportions may also be calculated using the approximation to the normal distribution and the sample standard deviation. The confidence interval for the population proportion may be calculated using the following equation: P% = p & z pq N where p is the proportion of successes, q is the proportion of failures (1 0 p), N is the sample size and z denotes the z value relating to a defined probability level. The use of this equation is described in the following example. EXAMPLE 6.8 A healthcare company that specialises in the sterile packaging of disposable gloves has decided to perform a quality audit of the resistance of a batch of packaged disposable gloves to the ingress of unsterile air. Out of a batch of 2500 packages, 200 were removed and

14 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals their sterility examined using the British Pharmacopoeia sterility tests. The results showed that 35 packages failed the sterility tests. On the basis of these observations, calculate the 95% confidence limits for the proportion of defective gloves in the batch (population). This problem may be solved using the equation described at the beginning of this section: P 95% = p & z 95% pq N where p # # 0.175, q # # 0.825, N # 200 and z # 1.96 (for a 95% probability, two-tailed). Thus: (0.175) (0.825) P 95% = & = & Therefore, the 95% confidence interval for the population proportion is , i.e. there is a 95% chance that the true proportion of defective gloves will fall within this range. Remember that the standard error of the proportion, ( pq N), may only be used in the calculation of the confidence intervals of a proportion if the sample size is sufficiently large for the normal approximation to be valid. As outlined in Chapter 4, the validity of this assumption may be rapidly estimated by calculating the product of the probability of the event and the sample size. Normality is commonly observed whenever this product exceeds Confidence intervals for differences between proportions The confidence intervals for differences in proportions may be estimated using a similar approach to that described in section 6.3, assuming that the proportions are obtained from a normal distribution. The main difference between the approach outlined in section 6.3 (for the calculation of confidence intervals for the difference between two means) and that required for the calculation of the confidence intervals for the differences in proportions is the mathematical definition of the standard error of the difference between means and proportions. Referring to section 6.3, it may be recalled that the standard error of the differences between two independent means is calculated using the following formula: SE (KX 1 KX 2 ) = s 2 1 N 1 + s 2 2 N 2

15 0894PP_ch06 15/3/02 11:03 am Page 149 Confidence intervals for differences between proportions 149 Conversely, the standard error of the difference between proportions is mathematically defined as: Applying the same mathematical approach as that described in section 6.3, the confidence intervals for the difference between two proportions may be defined as: where p 1 and p 2 are the two proportions under examination, z % is the z value corresponding to the accepted level of probability and SE diff is the standard error of the difference between proportions. The use (and calculation) of the confidence intervals relating to the difference between two proportions is explained in the following example. EXAMPLE 6.9 The relative efficacies of two oral antibiotics for the treatment of bronchial pneumonia have been assessed in a clinical trial. Subjects presenting with bronchial pneumonia at a hospital were divided into two groups, each of which received one or other of the antibiotics. The results from the clinical trial are shown in Table 6.1. Calculate the 95% confidence interval for the difference between the proportions of cured patients in the two groups. The equation used to calculate the 95% confidence interval for the difference between the two proportions is where SE ( p 1 p 2 ) = p 1 (1 p 1 ) + p 2 (1 p 2 ) = n 1 n 2 p 1 q 1 n 1 P% = ( p 1 p 2 ) & (z % SE diff ) = ( p 1 p 2 ) & z % p 1 (1 p 1 ) + p 2 (1 p 2 ) n 1 P 95% = ( p 1 p 2 ) & z 95% p 1(1 p 1 ) + p 2(1 p 2 ) n 1 p 1 = 575 ( ) = n 2 n 2 + p 2 q 2 n 2 Table 6.1 Cure Efficacy of two antibiotics for the treatment of bronchial pneumonia Number of patients Antibiotic 1 Antibiotic 2 Yes No

16 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals i.e. the proportion of cures associated with antibiotic 1; i.e. the proportion of cures associated with antibiotic 2; z 95% is 1.96 (95% probability, two-tailed outcome see Appendix 1); n 1 # (575! 45) # 620; and n 2 # (424! 200) # 625. When these values are substituted into the above equation, the confidence interval for the differences between the proportions of cures resulting from treatment with the two antibiotics may be calculated: Two further points should be mentioned at this stage. p 2 = 425 ( ) = P 95% = ( ) & ( ) 0.680( ) P 95% = & No units are associated with the calculated confidence interval of a difference in proportions, because proportions are, by definition, ratios. This is also true of the confidence intervals of a proportion. In this example, it has been assumed that the sample size is sufficiently large for the approximation to the normal distribution to apply. This may be easily confirmed by multiplying the number of observations in a particular group (e.g. n 1 ) by the probability (p 1 ). Values in excess of 6 confirm the validity of this assumption. The next example also describes the calculation of the confidence intervals associated with the difference between proportions. EXAMPLE 6.10 The effects of treatment of blastospores of Candida albicans (stationary phase) with either polyhexamethylenebiguanide (PHMB, 0.1% v v) or sterile water for 30 min on the number of blastospores adherent to buccal epithelial cells in vitro were examined. The results, expressed as the number of buccal epithelial cells with and without adherent blastospores, are presented in Table 6.2. Calculate the 95% confidence interval for the difference between the proportion of epithelial cells without adherent blastospores following treatment with either PHMB or sterile water. The relevant equation to calculate the 95% confidence intervals for the difference between the two proportions is as before: P 95% = ( p 1 p 2 ) & z 95% p 1 (1 p 1 ) + p 2 (1 p 2 ) n 1 n 2

17 0894PP_ch06 15/3/02 11:03 am Page 151 Confidence intervals and the t distribution 151 Table 6.2 Effects of treatment of blastospores of Candida albicans with either PHMB (0.1% v v) or sterile water on the subsequent adherence to buccal epithelial cells in vitro Outcome Treatment PHMB Water Cells without adherent blastospores Cells with adherent blastospores where p 1 # 103 (103! 48) # 0.682, i.e. the proportion of epithelial cells devoid of adherent blastospores following treatment with PHMB; p 2 # 54 (54! 91) # 0.372, i.e. the proportion of epithelial cells devoid of adherent blastospores following treatment with sterile water; z 95% is 1.96 (95% probability, two-tailed outcome, Appendix 1); n 1 # (103! 48) # 151; and n 2 # (54! 91) # 145. Substitution of these values into the above equation and computation yields the 95% confidence interval, i.e & 0.11 # It may sometimes be difficult to fully appreciate the meaning of the calculated confidence interval for the difference between two proportions. Therefore, in medical research, it is commonplace to express the difference between proportions as ratios. In Example 6.10 it may be observed that treatment of blastospores with PHMB is more likely to inhibit adherence to buccal epithelial cells by a factor of # An alternative expression that is commonly employed to relate two ratios is the odds ratio. The odds of an event may be defined as the ratio of the probability of the occurrence of an event to that of the non-occurrence of the event. Reverting to Example 6.10, the odds of observing buccal epithelial cells devoid of adherent blastospores following treatment with PHMB is (0.682 ( ) # Similarly, the odds of observing a buccal epithelial cell devoid of adherent blastospores following treatment with sterile water are (0.373 ( ). The ratio of these two odds is termed the odds ratio or relative risk, which for the example described, is # Confidence intervals and the t distribution As highlighted in the introduction to this chapter, the z statistic is employed to calculate confidence intervals whenever the assumptions of the normal distribution are valid. One requirement is a large sample size. However, the inquisitive reader may ask, how are confidence intervals calculated for sample data? Under these circumstances, the t distribution is employed to calculate confidence intervals, using a similar

18 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals strategy to that used to calculate the confidence intervals of the true mean for large samples. Thus: ts P% = KX & N where P% is the chosen probability (usually 90%, 95% or 99%). KX is the sample mean, t is the t statistic corresponding to the appropriate probability level and degree of freedom and S is the sample standard deviation The major difference between the above equation and the equation for the calculation of the confidence interval using the normal distribution is the nature of the test statistic. In the normal distribution the value of the z statistic depends only on the chosen probability level and is independent of sample size (the use of the normal distribution implies that the sample size is large). The t statistic, however, is dependent on two parameters: the chosen probability and, additionally, the number of degrees of freedom, which is itself dependent on the size of the sample (df # N 0 1). It is important that the reader fully understands the nature of the differences between these two methods of calculating the confidence intervals of the population mean. The use of the t distribution to calculate confidence intervals of the population mean is illustrated in the following examples. EXAMPLE 6.11 Table 6.3 shows the concentration of chlorhexidine (mg) in 15 1-mL samples that have been removed from a stock solution (5 L) and assayed by ultraviolet spectroscopy. Calculate the 95% confidence interval of the population mean. First, calculate the mean and standard deviation, i.e & 0.01 mg ml. As the sample size is small (N # 15), the confidence intervals must be calculated using the t distribution, i.e. ts P 95% = KX & N The mean and standard deviation have been calculated using the sample data. The final unknown in the equation is therefore the t statistic. Two Table 6.3 Concentration of chlorhexidine (mg ml) in 15 samples removed from a stock solution

19 0894PP_ch06 15/3/02 11:03 am Page 153 Confidence intervals and the t distribution 153 points should be considered before selecting this value from Appendix 2: the probability level (95%, two-tailed) and the number of degrees of freedom. The latter is calculated by subtracting 1 from the sample size, i.e # 14 df. The t statistic corresponding to 14 df and 95% probability is 2.14 (Appendix 2). The confidence intervals may then be calculated: The confidence interval is therefore mg ml. A second example of this calculation is provided below. EXAMPLE 6.12 A chemist has developed a synthetic route for the preparation of a compound with antihistamine properties. A 10-kg batch has been prepared, from which mg samples have been removed and their purity determined using an analytical method. The results are shown in Table 6.4. Calculate the 95% and 99% confidence intervals of the true purity of the compound. The first stage of the calculation involves the calculation of the mean and standard deviation of the data described in Table 6.4, i.e & 2.56%. ts 0.01 P 95% = KX & = & 2.14 N 15 95% confidence interval ts P 95% = KX & N = & mg ml where the mean and standard deviation have been determined as and 2.56% and N, the sample size, is 10. The relevant value of the t statistic is derived from the table of the critical values of the t distribution (Appendix 2) with knowledge of the number of degrees of freedom ( # 9) and the chosen level of probability (95%, two-tailed outcome). The t statistic is therefore Consequently, the 95% confidence interval is ts 2.56) P 95% = KX & = & (2.26 = & 1.83% N 10 99% confidence interval. As the probability associated with the confidence interval has changed, the t statistic must be modified. From Appendix 2, the t Table 6.4 Purities (%) of 10 aliquots of an antihistamine removed from a single batch of product

20 0894PP_ch06 15/3/02 11:03 am Page Statistical estimation using confidence intervals statistic associated with 9 df, two-tailed outcome and 99% probability is This knowledge allows calculation of the 99% confidence interval: ts 2.56) P 95% = KX & = & (3.25 = & 2.63% N Conclusions In this chapter the theory and applications of confidence intervals have been outlined. It is important to understand the concept of confidence intervals, because they are frequently employed to provide information about the magnitude of the true statistic from sample data. In particular, the confidence intervals associated with several descriptive statistics have been described, namely the mean, standard deviation, proportions and differences between these parameters. Furthermore, the uses of both the z and t statistics in the calculation of confidence intervals have been illustrated, and indeed this is the first example of the use of the t distribution for the statistical characterisation of small samples. Further examples of this application will be described in subsequent chapters.

### Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

### 99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm

Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the

### 5.1 Identifying the Target Parameter

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

### Sampling Distributions and the Central Limit Theorem

135 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 10 Sampling Distributions and the Central Limit Theorem In the previous chapter we explained

### Means, standard deviations and. and standard errors

CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### 10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

### Content Sheet 7-1: Overview of Quality Control for Quantitative Tests

Content Sheet 7-1: Overview of Quality Control for Quantitative Tests Role in quality management system Quality Control (QC) is a component of process control, and is a major element of the quality management

### Size of a study. Chapter 15

Size of a study 15.1 Introduction It is important to ensure at the design stage that the proposed number of subjects to be recruited into any study will be appropriate to answer the main objective(s) of

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

### Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1: THE NORMAL CURVE AND "Z" SCORES: The Normal Curve: The "Normal" curve is a mathematical abstraction which conveniently

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### (Refer Slide Time: 00:00:56 min)

Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide

### Sampling and Hypothesis Testing

Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus

### find confidence interval for a population mean when the population standard deviation is KNOWN Understand the new distribution the t-distribution

Section 8.3 1 Estimating a Population Mean Topics find confidence interval for a population mean when the population standard deviation is KNOWN find confidence interval for a population mean when the

### Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

### CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

7 Statistical Intervals Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### CHAPTER THREE COMMON DESCRIPTIVE STATISTICS COMMON DESCRIPTIVE STATISTICS / 13

COMMON DESCRIPTIVE STATISTICS / 13 CHAPTER THREE COMMON DESCRIPTIVE STATISTICS The analysis of data begins with descriptive statistics such as the mean, median, mode, range, standard deviation, variance,

### In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing.

3.0 - Chapter Introduction In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing. Categories of Statistics. Statistics is

### Hence, multiplying by 12, the 95% interval for the hourly rate is (965, 1435)

Confidence Intervals for Poisson data For an observation from a Poisson distribution, we have σ 2 = λ. If we observe r events, then our estimate ˆλ = r : N(λ, λ) If r is bigger than 20, we can use this

### Chapter 7. Estimates and Sample Size

Chapter 7. Estimates and Sample Size Chapter Problem: How do we interpret a poll about global warming? Pew Research Center Poll: From what you ve read and heard, is there a solid evidence that the average

### Report of for Chapter 2 pretest

Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every

### DATA INTERPRETATION AND STATISTICS

PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### The normal approximation to the binomial

The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

### How to Verify Performance Specifications

How to Verify Performance Specifications VERIFICATION OF PERFORMANCE SPECIFICATIONS In 2003, the Centers for Medicare and Medicaid Services (CMS) updated the CLIA 88 regulations. As a result of the updated

### Outline. 1 Confidence Intervals for Proportions. 2 Sample Sizes for Proportions. 3 Student s t-distribution. 4 Confidence Intervals without σ

Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for Proportions 3 Student s t-distribution 4 Confidence Intervals without σ Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for

### Table 2-1. Sucrose concentration (% fresh wt.) of 100 sugar beet roots. Beet No. % Sucrose. Beet No.

Chapter 2. DATA EXPLORATION AND SUMMARIZATION 2.1 Frequency Distributions Commonly, people refer to a population as the number of individuals in a city or county, for example, all the people in California.

### Standard Deviation Calculator

CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

### 3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

### 6. Methods 6.8. Methods related to outputs, Introduction

6. Methods 6.8. Methods related to outputs, Introduction In order to present the outcomes of statistical data collections to the users in a manner most users can easily understand, a variety of statistical

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

### MEASURES OF VARIATION

NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

### 9Tests of Hypotheses. for a Single Sample CHAPTER OUTLINE

9Tests of Hypotheses for a Single Sample CHAPTER OUTLINE 9-1 HYPOTHESIS TESTING 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical Hypotheses 9-1.3 One-Sided and Two-Sided Hypotheses 9-1.4 General

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Standard Deviation Estimator

CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### 2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

### Data Analysis: Describing Data - Descriptive Statistics

WHAT IT IS Return to Table of ontents Descriptive statistics include the numbers, tables, charts, and graphs used to describe, organize, summarize, and present raw data. Descriptive statistics are most

### IV solutions may be given either as a bolus dose or infused slowly through a vein into the plasma at a constant or zero-order rate.

د.شيماء Biopharmaceutics INTRAVENOUS INFUSION: IV solutions may be given either as a bolus dose or infused slowly through a vein into the plasma at a constant or zero-order rate. The main advantage for

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### GCSE Statistics Revision notes

GCSE Statistics Revision notes Collecting data Sample This is when data is collected from part of the population. There are different methods for sampling Random sampling, Stratified sampling, Systematic

### A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

### 2015 Junior Certificate Higher Level Official Sample Paper 1

2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### Estimating Weighing Uncertainty From Balance Data Sheet Specifications

Estimating Weighing Uncertainty From Balance Data Sheet Specifications Sources Of Measurement Deviations And Uncertainties Determination Of The Combined Measurement Bias Estimation Of The Combined Measurement

### Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

### Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

### Statistical Confidence Calculations

Statistical Confidence Calculations Statistical Methodology Omniture Test&Target utilizes standard statistics to calculate confidence, confidence intervals, and lift for each campaign. The student s T

### Descriptive Statistics

Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

### 11. Analysis of Case-control Studies Logistic Regression

Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Hypothesis testing for µ:

University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative

### OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance

Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance (if the Gauss-Markov

### Sample Size Determination

Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

### Simple Random Sampling

Source: Frerichs, R.R. Rapid Surveys (unpublished), 2008. NOT FOR COMMERCIAL DISTRIBUTION 3 Simple Random Sampling 3.1 INTRODUCTION Everyone mentions simple random sampling, but few use this method for

### The correlation coefficient

The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### How do I analyse observer variation studies?

How do I analyse observer variation studies? This is based on work done for a long term project by Doug Altman and Martin Bland. I recommend that you first read our three Statistics Notes on measurement

### F. Farrokhyar, MPhil, PhD, PDoc

Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

### 5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0

a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

### Using simulation to calculate the NPV of a project

Using simulation to calculate the NPV of a project Marius Holtan Onward Inc. 5/31/2002 Monte Carlo simulation is fast becoming the technology of choice for evaluating and analyzing assets, be it pure financial

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Guidance for Industry

Guidance for Industry Q2B Validation of Analytical Procedures: Methodology November 1996 ICH Guidance for Industry Q2B Validation of Analytical Procedures: Methodology Additional copies are available from:

### CHINHOYI UNIVERSITY OF TECHNOLOGY

CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 2 Simple Linear Regression

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 2 Simple Linear Regression Hi, this is my second lecture in module one and on simple

### Understanding Process Capability Indices

Understanding Process Capability Indices Stefan Steiner, Bovas Abraham and Jock MacKay Institute for Improvement of Quality and Productivity Department of Statistics and Actuarial Science University of

### Chapter 8 Design of Concrete Mixes

Chapter 8 Design of Concrete Mixes 1 The basic procedure for mix design is applicable to concrete for most purposes including pavements. Concrete mixes should meet; Workability (slump/vebe) Compressive

### Point and Interval Estimates

Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

### GUIDELINES FOR THE VALIDATION OF ANALYTICAL METHODS FOR ACTIVE CONSTITUENT, AGRICULTURAL AND VETERINARY CHEMICAL PRODUCTS.

GUIDELINES FOR THE VALIDATION OF ANALYTICAL METHODS FOR ACTIVE CONSTITUENT, AGRICULTURAL AND VETERINARY CHEMICAL PRODUCTS October 2004 APVMA PO Box E240 KINGSTON 2604 AUSTRALIA http://www.apvma.gov.au

### Confidence Intervals for Cp

Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process

### CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES

CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES The chi-square distribution was discussed in Chapter 4. We now turn to some applications of this distribution. As previously discussed, chi-square is

### HOW ACCURATE ARE THOSE THERMOCOUPLES?

HOW ACCURATE ARE THOSE THERMOCOUPLES? Deggary N. Priest Priest & Associates Consulting, LLC INTRODUCTION Inevitably, during any QC Audit of the Laboratory s calibration procedures, the question of thermocouple

### Statistics Summary (prepared by Xuan (Tappy) He)

Statistics Summary (prepared by Xuan (Tappy) He) Statistics is the practice of collecting and analyzing data. The analysis of statistics is important for decision making in events where there are uncertainties.

### 18.6.1 Terms concerned with internal quality control procedures

18.6.1 Terms concerned with internal quality control procedures Quality assurance in analytical laboratories Quality assurance is the essential organisational infrastructure that underlies all reliable

### Measurement & Lab Equipment

Measurement & Lab Equipment Materials: Per Lab Bench (For the following materials, one of each per bench, except for rulers.) Thermometers 6 Metric Rulers - 6 or 12 School Rulers 12 Digital Balance - 6

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution