Chapter 11. Inference For Distribution

Transcription

1 Chapter 11 Inference For Distribution

2 Introduction Single Population when σ unknown Confidence Intervals Significance Test Comparing two populations when σ unknown Confidence Intervals Significance Test

3 Lesson 11-1, Part 1 Inference for the Mean of a Population

4 Conditions for Testing claims about the Population Mean σ Unknown Simple random sample (SRS) Distribution is approximately normal if Population is normally distribution The sampling distribution will resemble that of the population. n 30 The Central Limit Theorem tells us if the sampling distribution of the sample mean will be approximately normal.

5 Conditions for Testing claims about the Population Mean σ Unknown n < 30 Normal probability plot show a linear trend with no outliers.

6 One-Sample t Statistic z x t x s n n standard error of the sample mean is s n

7 Density Curve for the t-distribution Symmetric about zero, single peaked, and bell shape The spread is a bit greater than a normal distribution Has more probability in the tails and less in the center As the degree of freedom (n 1) increases it becomes approximately normal.

8 t Confidence Interval z confidence interval t confidence interval x z* n x t * s n A confidence interval or significance test is called robust if the confidence level or P value does not change very much when the assumptions are violated.

9 Example Page 620, #11.2 What are the critical value t* from Table C satisfies each of the following conditions? A). the t distribution with 5 degrees of freedom has probability 0.05 to the right of t*

10 Example Page 620, #11.2 B). the t distribution with 21 degrees of freedom has probability 0.99 to the left of t*

11 Example Page 620, #11.4 What is the critical value t* from table C should be used for a confidence interval for the mean of the population in each of the following situations? A) A 90% confidence interval based on n = 12 observations. df n

12 Example Page 620, #11.4 B) A 95% confidence interval from an SRS of 30 observations. df n

13 Example Cricket Love The article Well Fed Crickets Bowl of Maidens Over (Nature Science Update, February 11, 1999) reported that female field crickets are attracted to males that have high chirp rates and hypothesized that chirp rate is related to nutritional status. The usual chirp rate for male field crickets was reported to vary around a mean of 60 chirps per second. To investigate whether chirp rate was related to nutritional status, investigators fed male crickets a high protein diet for 8 days, after which chirp rate was measured. The mean chirp rate for the crickets on the high protein diet was reported at 109 chirps per second. Is this convincing evidence that the mean chirp rate for crickets on a high protein diet is greater than 60 (which would then apply an advantage in attracting the ladies)? Suppose that the sample size and sample standard deviation are n = 32, s = 40. We test the relevant hypothesis with = 0.01.

14 Example Cricket Love Step 1 Identify the population of interest and parameter you want to draw conclusions about. μ = mean chirp rate for crickets on a high protein diet. H H o a : 60 : 60

15 Example Cricket Love Step 2 Identify the statistical procedure you should use. Then verify the conditions for using this procedure. use a one sample t test with the following conditions: Simple random sample (assume) Sampling distribution is approximately normal since n 30.

16 Example Cricket Love Step 3 Carry out the inference procedure. t x o s n P value df 31 0

17 Example Cricket Love Step 4 Interpret your results in the context of the problem There is sufficient evidence to reject H o since P-value = 0 =.01 and conclude that the mean chirp rate is higher for male crickets that eat a higher protein diet.

18 Example Page 628, #11.10 Here are estimates of the daily intakes of calcium (in milligrams) for 38 women between the ages of 18 and 24 years who participated in a study of women s bone health:

19 Example Page 628, #11.10 A). Display the data using a stemplot and make a normal probability plot. Describe the distribution of calcium intakes for these women nd Y= Plot

20 Example Page 628, # The data are right skewed with a high outlier of 2433 and possible 1933.

21 Example Page 628, #11.10 B). Calculate the mean, the standard deviation, and the standard error. x s s SE 69.3 n 38

22 Example Page 628, #11.10 C). Find a 95% confidence interval for the mean. Use the inference toolbox. Step 1 Identify the population of interest and the parameter you want to draw conclusion about. μ =, the mean daily intakes of calcium for women between that ages of 18 and 24.

23 Example Page 628, #11.10 Step 2 Identify the statistical procedure you should use. Then verify the conditions for using this procedure. Use a one-sample t interval. Conditions 1. Simple random sample - assume 2. The sampling distribution is approximately normal since n 30.

24 Example Page 628, #11.10 Step 3 Carry out the inference procedure. x s t* n 38 STAT TESTS TInterval to use df = 37 use 30 in Table C

25 Example Page 628, #11.10 Step 4 Interpret your results in the context of the problem We are 95% confident that the true mean daily intake of calcium for women between is 18 and 24 years is between mg and mg.

26 Example Page 628, #11.10 D) Eliminate the two largest values and recompute the 95% confidence interval. What do you notice? The normal quantile plot is essentially the same as before (except the two points that deviated greatly from the line are gone).

27 Lesson 11-1, Part 2 Inference for the Mean of a Population

28 Matched Pairs To compare the responses to the two treatments in a matched pair, apply the one-sample t procedure to the observed differences.

29 Example Page 643, #11.29 The researchers studying vitamin C in CSB in Exercise 11.9 (page 628) were also interested in similar commodity called wheat soy blend (WSB). A major concern was the possibility that some of the vitamin C content would be destroyed as a result of storage and shipment of the commodity to its final destination. The researchers specially marked a collection of bags at the factory and took a sample form each of these to determine the vitamin C content. Five months later in Haiti they found the specially marked bags and took samples. The consist of two vitamin C measures for each bag, one at the time of production in the factory and the other five months later in Haiti. The units are mg/100g.

30 Example Page 643, #11.29 Factory Haiti Factory Haiti Factory Haiti

31 Example Page 643, #11.29 A). Examine the question of interest to these researchers. Provide appropriate statistical evidence to justify you conclusions. Step 1 Identify the population of interest and the parameter you want to draw conclusion about. µ = mean change (Haiti factory) of vitamin C in mg/100g. H H o a : 0 : 0

32 Example Page 643, #11.29 Step 2 Identify the statistical procedure you should use. Then verify the conditions for using this procedure. We will use a one-sample t test. Conditions 1. Simple random sample - assumed 2. The sampling distribution is approximately normal since the normal probability plot also appears to reasonably linear with no outliers.

33 Example Page 643, #

34 Example Page 643, #11.29 Step 3 Calculate the test statistic and p-value. Illustrate with a graph. t x diff s diff n o STAT TESTS T-Test p value df 35

35 Example Page 643, #11.29 Step 4 Interpret your results in the context of the problem There is sufficient evidence to reject H o (P-value = < α = 0.01) and conclude that the mean change between Haiti and factory is less than 0. This indicates that the average amount of vitamin C has decrease as a result of storage and shipment.

36 Example Page 643, #11.29 B). Estimate the loss in vitamin C content over the five-month period. Use a 95% confidence level. Step 3 Carry out the inference procedure. x diff t s * diff n ( 7.544, 3.123) We are 95% confident that the population mean loss in vitamin C content over a five month period is between mg/100g and mg/100g. STAT TESTS TInterval t* use table C with a df = 26

37 Example Page 643, #11.29 C). Do these data provide evidence that the mean vitamin C content of all bags of WSB shipped to Haiti differs from the target value of 40 mg/100g. Step 1 Identify the population of interest and the parameter you want to draw conclusion about. μ = mean of vitamin C in mg/100g for all bags shipped to Haiti. H H o a : 40 : 40

38 Example Page 643, #11.29 Step 3 Calculate the test statistic and p-value. Illustrate with a graph. t x o s n STAT TESTS T-Test p value df

39 Example Page 643, #11.29 Step 4 Interpret your results in the context of the problem There is sufficient evidence to reject H o (P-value = < α = 0.01 and conclude that the mean vitamin C content is different from the target mean of 40 mg/100g.

40 Robustness of t Procedures The t procedures are strongly influenced by outliers. Always check the data first! If there are outliers and the sample size is small, the results will not be reliable. The t procedures are robust when there are no outliers, especially when the distribution is approximately symmetric.

41 When to Use t Procedures If the sample size is less than 15, only use t procedures if the data are close to normal. If the sample size is at least 15, only use t procedures if there are no outliers. If the sample size is at least 40, you may use t procedures, even if the data is skewed.

42 Lesson 11-2 Comparing Two Sample Means

43 Two-Sample Problems In a comparative study, we want to compare the responses to two treatments or to compare the characteristic of two populations. We have separate samples for each treatment or each population. The samples must be chosen randomly and independently in order to perform statistical inference. Because matched pairs are NOT chosen independently, we will NOT use two-sample inference for a matched pair design. For a matched pair design, apply the one-sample t procedures to observe difference.

44 Notation for Two Samples Population Variable Mean Standard Deviation 1 x 1 μ 1 σ 1 2 x 2 μ 2 σ 2 Population Sample Size Sample Mean Sample Standard Deviation 1 n 1 s 1 2 n 2 s 2 x 1 x 2

45 Null and Alternative Hypothesis The null hypothesis is that there is no difference between the two parameters. H o : µ 1 = µ 2 or H o : µ 1 µ 2 = 0 The alternative hypothesis could be that H a : µ 1 µ 2 (two-sided) H a : µ 1 < µ 2 or H a : µ 1 µ 2 < 0 (one-sided) H a : µ 1 > µ 2 or H a : µ 1 µ 2 > 0 (one-sided)

46 Conditions for Comparing Two Samples Two random samples (SRS) that are independent. Both populations are normally distributed if For each sample either population is normal distributed The sampling distribution will resemble that of the population.

47 Conditions for Comparing Two Samples n 1 30 and n 2 30 The central limit theorem tells us the sampling distribution of the mean will be approximately normal. n 1 < 30 and n 2 < 30 The sampling distribution of the mean will be approximately normal if the normal probability plot is linear with no outliers.

48 Sampling Distribution of x x 1 2 Mean Standardize z μ μ 1 2 Variance σ n 2 σ n 1 2 z z x σ μ x x μ μ σ 2 σ n n 1 2

49 Test Statistic for Two Samples when σ Unknown t x x s s 1 2 n n 1 2 Standard Error Estimated Standard Deviation

50 Confidence Interval for Two Samples when σ Unknown Confidence interval for μ 1 μ 2 given by 2 2 s s 1 2 x x t * 1 2 n n 1 2 t* 1 C 2 Degree of freedom is equal to the smaller of n 1 1 or n 2 1.

51 Example Page 657, #11.40 How badly does logging damage tropical rainforests? One study compared forest plots in Borneo that had never been logged with similar plots nearby that had been logged 8 years earlier. The study found that the effects of logging were somewhat less severe than expected. Here are the data on the number of tree species in 12 unlogged plots and 9 logged plots: Unlogged: Logged:

52 Example Page 657, #11.40 A) The study report says, Loggers were unaware that the effects of logging would be assessed. Why is this important? The study report also explains why the plots can considered to be randomly assigned. If the loggers had known that a study would be done, they might have (consciously or subconsciously) cut down fewer trees than they typically would, in order to reduce the impact of logging.

53 Example Page 657, #11.40 B) Does logging significantly reduce the mean number of species in a plot after 8 years. Give appropriate statistical evidence to support your conclusion. We want to compare the mean number tree species of unlogged (μ 1 ) plots versus logged (μ 2 ) plots. H H o a : 1 2 : 1 2 H H : 0 : 0 o 1 2 or a 1 2

54 Example Page 657, #11.40 We will use a two sample test Conditions: 1. Plots are randomly assigned, not sure if it s a SRS so we may be able to generalize the results 2. The stemleaf plot and the normal probability plot appear to be approximately normal.

55 Example Page 657, #11.40 Unlogged Logged

56 Example Page 657, #11.40 STAT TESTS 2-SampTTest

57 Example Page 657, #11.40 t x x s s n n 1 2 t ( ) (0) P-value = df =

58 Example Page 657, #11.40 There is sufficient evidence to reject H o (P-value = < α = 0.05) and conclude that logging reduce the mean number of tree species. Also note that this would be not be statistical significant at 1%.

59 Example Page 657, #11.40 C). Give a 90% confidence interval for the difference in the mean number of species between unlogged and logged plots. STAT TESTS 2-SampTInt

60 Example Page 657, #11.40 s x 1 x2 t * n 2 2 s 1 2 n ( ) (0.6517, 7.015) t* = where df = 14.79