MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for the following: 1. Proportion (percentage of population that possesses some characteristic or trait) 2. Mean (average value of whatever quantity is being measured for the population) 3. Variance (NOT standard deviation, although variance is used to work with standard deviation) A confidence interval (CI) for one of the parameters listed above is generally constructed through the following process: 1. Determine the corresponding statistic from a sample 2. Determine the range of values (upper and lower bounds) assumed to contain the true value of the population parameter at the stated level of confidence It should be noted that although following the above procedure is the general process for making inferences about a population parameter based on a sample statistic, each of the three parameters for which valid inferences can be made has its own unique formulas to use. Furthermore, obtaining the range of values that is hoped to contain the true value of the actual population parameter does not guarantee that the true value of the population parameter actually falls in that range. Related to confidence intervals is the concept of determining the minimum sample size necessary to ensure valid results, which is done simply by solving the pertinent formula for where applicable. The following variables and abbreviations are used at various times when performing calculations related to confidence intervals and, later on, hypothesis testing (which is covered in a future chapter): = population proportion ( is the complement of ; and ) = population mean = population variance = population standard deviation = sample proportion ( is the complement of ; and ) = sample mean = sample variance = sample standard deviation = sample size = confidence interval = margin of error (used for calculating for or ) = complement of confidence level associated with (for example, a confidence level means, which means ) = area in each tail for a confidence interval ( represents the middle range of values) = critical value in normal distribution for for or with known value of = critical value in distribution for for with unknown value of = degrees of freedom ( ) = critical value in distribution for lower bound of for (or ) = critical value in distribution for upper bound of for (or )
Procedure for Estimating a Population Proportion b. The conditions for binomial distribution are satisfied: i. Fixed number of trials ( is set) ii. Trials are independent (the outcome of any one trial has no effect on any other trial) iii. Exactly two categories of outcomes for each trial (success and failure) iv. Probabilities remain constant for each trial ( and do not change) c. There are at least successes and failures (equivalent to and ) 2. If not provided, calculate the value of the sample proportion : (Note: 4. Look up the value of in the table of values for the normal distribution 5. Calculate the margin of error : 6. Calculate the limits and state the for the population proportion in one of the following ways: a. Preferred method: b. c. ( of ( and margin of error, simply solve the equation for above for : [ ] Notes: 1. It is highly unlikely that there will be prior knowledge to provide a value for. In such instances, it is customary to use (and, therefore, ) in the calculation for. (If prior knowledge provides a value of to be used, then that should be used instead.) 2. Always round the value of up to the next highest integer if the value is not an integer.
Procedure for Estimating a Population Mean When Is Known Note: In general, this procedure is not used since it would be extremely unlikely that a value for would be known without also knowing the true value of since the calculation of depends on the value of, although rare cases may exist; the more practical approach for estimating is outlined in the next section of this packet. b. The value of the population standard deviation is known c. Either or both of the following conditions are met: i. The population is normally distributed ii. The sample size 2. If not provided, calculate the value of the sample mean 4. Look up the value of in the table of values for the normal distribution 5. Calculate the margin of error : 6. Calculate the limits and state the for the population mean in one of the following ways: a. Preferred method: b. c. ( of ( and margin of error, simply solve the equation for above for : [ ] Note: Always round the value of up to the next highest integer if the value is not an integer.
Procedure for Estimating a Population Mean When Is Not Known b. Either or both of the following conditions are met: i. The population is normally distributed ii. The sample size 2. If not provided, calculate the value of the sample mean 3. If not provided, calculate the value of the sample standard deviation 4. Determine the value of based on the stated confidence level: 5. Determine the number of degrees of freedom, : 6. Look up the value of in the table of values for the distribution 7. Calculate the margin of error : 8. Calculate the limits and state the for the population mean in one of the following ways: a. Preferred method: b. c. ( of ( and margin of error, it is generally acceptable to simply solve the equation for above for and use the value in the row labeled Large for degrees of freedom as the value of (this is equivalent to the value of used in the formula from the previous section in which the population standard deviation is known): [ ] Notes: 1. If statistics from a sample of known size are provided, the value of from the row for the corresponding number of degrees of freedom can be used, resulting in a slightly larger minimum necessary sample size which will theoretically provide more accurate results. 2. To obtain the greatest minimum necessary sample size using this approach, the preceding formula can be used with degree of freedom, although the resulting sample size will be significantly larger and may not be practical when formally gathering data. 3. If the population cannot be assumed to be at least reasonably close to normally distributed, calculating a minimum sample size becomes complex beyond the scope of this course. 4. Always round the value of up to the next highest integer if the value is not an integer.
Procedure for Estimating a Population Variance b. The population must be normally distributed regardless of sample size 2. If not provided, calculate the value of the sample standard deviation 4. Determine the number of degrees of freedom, : 5. Look up the values of and in the table of values for the distribution 6. Calculate the limits and state the for the population variance : ( ( 7. If the for the population standard deviation is desired, simply take square roots: ( ( Because the distribution depends on the number of degrees of freedom, and the number of degrees of freedom depends on the sample size, it is not possible to determine the minimum sample size necessary to guarantee valid results for a stated confidence level of ( and margin of error by simply solving the equation for above for. However, complex procedures do exist to determine such sample sizes. Since those procedures are beyond the scope of this course, it is sufficient to simply note that Table 7-2 on page 376 of the textbook provides required minimum sample sizes for many of the most common confidence levels and margins of error. When necessary throughout this course, it is sufficient to simply look up a required minimum sample size in that table.