10. exercises_Means

Introductory Statistics Explained (1.11) Exercises Inference for Two Means © 2022, 2023, 2024 Jeremy Balka Comparing Two Means v1.11 W24 Draft J.B.’s strongly suggested exercises: 4 , 6 , 7 , 9 , 10 , 11 , 17 , 23 , 26 , 27 , 30 , 31 , 33 NB The section titles and numbers are not yet synced up with the text. 1 Introduction 2 The Sampling Distribution of the Di ff erence in Sample Means 1. Suppose we draw a random sample from Population 1, which has a mean of 15 and a standard deviation of 2 , and we draw a random sample from Population 2, which has a mean of 8 and a standard deviation of 6 , and the two samples can be considered independent. (a) If n 1 = n 2 = 4 , what is the mean of the sampling distribution of ¯ X 1 - ¯ X 2 ? (b) If n 1 = n 2 = 4 , what is the standard deviation of the sampling distribution of ¯ X 1 - ¯ X 2 ? (c) If n 1 = n 2 = 4 , what is the shape of sampling distribution of ¯ X 1 - ¯ X 2 ? (d) If n 1 = n 2 = 400 , what is the mean of the sampling distribution of ¯ X 1 - ¯ X 2 ? (e) If n 1 = n 2 = 400 , what is the standard deviation of the sampling distribution of ¯ X 1 - ¯ X 2 ? (f) If n 1 = n 2 = 400 , what is the shape of sampling distribution of ¯ X 1 - ¯ X 2 ? 2. Tom and Pete are two NFL prospects at the NFL combine, where players eligible for the NFL draft show their skills in a variety of di ff erent exercises. Tom and Pete are about to have three attempts at the 40 yard dash. Suppose that (theoretically) Tom’s times in this event are approximately normally distributed with a mean of 4.612 seconds and a standard deviation of 0.048 seconds. Pete’s times are approximately normally distributed with a mean of 4.528 seconds and a standard deviation of 0.044 seconds. Suppose it is reasonable to assume independence between runs. (The runs may not be truly independent, but this assumption provides a reasonable approximate model.) (a) In their first attempt, what is the probability that Tom’s time is greater than Pete’s time? (b) What is the probability that Tom’s average time in the three attempts is greater than Pete’s average time in the three attempts? 1

3 Hypothesis Tests and Confidence Intervals for Two Indepen- dent Samples (When σ 1 and σ 2 are known) 4 Hypothesis Tests and Confidence Intervals for μ 1 - μ 2 (When σ 1 and σ 2 are unknown) 4.1 Pooled-Variance t Tests and Confidence Intervals 3. Table 1 illustrates the results of a two-sample study. The samples were drawn independently from normally distributed populations. Group 1 Group 2 Sample mean 8.8 17.2 Sample standard deviation 1.42 2.61 Sample size 10 5 Table 1: Results of a two-sample study. For the pooled-variance t procedure, calculate: (a) The point estimate of μ 1 - μ 2 . (b) The pooled variance s 2 p . (c) SE ( ¯ X 1 - ¯ X 2 ) . (d) A 95% confidence interval for μ 1 - μ 2 . (e) Test the null hypothesis that the population means are equal. Give the appropriate hypotheses, standard error, value of the test statistic, p -value, and conclusion at ↵ = 0 . 05 . 4. Does a vitamin D supplement have an e ff ect on parathyroid hormone (PTH) levels in the blood? An experiment 1 investigated the e ff ect of a vitamin D supplement on several biological factors in study participants. One of the variables was the change in PTH in the blood. In the experiment, 26 individuals were randomly assigned to one of two groups. Each group consumed 240 mL of orange juice per day for 12 weeks. The orange juice of the treatment group was fortified with 1000 IU vitamin D 3 , whereas the control group’s orange juice had no vitamin D 3 added. After 12 weeks, the change in PTH level in the blood (pg/mL) was recorded. Table 4 illustrate the results. Fortified with vitamin D ¯ X 1 = - 9 . 0 s 1 = 37 . 5 n 1 = 14 Not fortified with vitamin D ¯ X 2 = - 1 . 6 s 2 = 34 . 6 n 2 = 12 Table 2: Change in blood PTH levels after 12 weeks. Suppose that normal quantile-quantile plots showed that the values were approximately normally distributed. Use the pooled-variance t procedure to answer the following questions. (Since the observations are approximately normally distributed, and the sample standard deviations are similar, the pooled-variance t procedure is a reasonable method of analysis.) 1 Tangpricha et al. (2003). Fortification of orange juice with vitamin D: a novel approach for enhancing vitamin D nutritional health. The American journal of clinical nutrition , 77(6): 1478-1483. Ho M M Ha u do

Welch Two Sample t-test data: vitaminD and novitaminD t = -1.2156, df = 13.215, p-value = 0.2454 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -16.923691 4.723691 sample estimates: mean of x mean of y -6.3 -0.2 Give a summary of the results of the analysis. 4.3 Guidelines for Choosing the Appropriate Two-Sample t Procedure 7. What factors influence the choice between the Welch procedure and the pooled-variance t procedure? When is the Welch procedure a better choice? When is the pooled-variance t procedure a better choice? 8. Suppose we are interested in assessing the e ff ect of two fuel additives on fuel e ffi ciency. We intend to use a t procedure to carry out an appropriate hypothesis test. Under which of the following situations would it be most appropriate to use Welch’s approximation instead of the pooled-variance procedure? (a) The sample sizes of the two groups are similar, and the sample standard deviations are similar. (b) The sample sizes of the two groups are similar, and the sample standard deviations are very di ff erent. (c) The sample sizes of the two groups are very di ff erent, and the sample standard deviations are similar. (d) The sample sizes of the two groups are very di ff erent, and the sample standard deviations are very di ff erent. 5 Paired-Di ff erence Procedures 5.1 Paired-Di ff erence t Tests and Confidence Intervals 9. O ff erman et al. ( 2009 ) conducted an experiment on pigs to investigate the e ff ect of an antivenom after an injection of rattlesnake venom. In one aspect of the study, the researchers investigated the change in volume of the right hind leg before and after being subjected to a dose of venom and treatment with an antivenom. The volume of the right hind leg was measured in 9 pigs before being injected with the venom, then the pigs were injected with a dose of venom and treated intravenously with an antivenom, and after 8 hours the volume of the leg was measured again. The results are illustrated in Table 5 . (The volume was measured using a water displacement method; the units are mL.) (a) Suppose we are interested in estimating the true mean amount of swelling for pigs of this type under the conditions of this experiment. Would we use a paired-di ff erence procedure or an independent sample procedure? Justify your response. (b) What are the 9 di ff erences? (Take the di ff erences as After - Before.) (c) What is the sample mean di ff erence? What is the standard deviation of the di ff erences? (d) What is the standard error of the sample mean di ff erence? large p rake so no strong evidenc errett of diff variances when sample size r diff so pooled not good paired diff bic2 samples on same unit before after Not indep x ̅ say 203.3 IN R data pig SE 5 8 18.7268 mean pig 7203.333 sd pig i 56.180s

Before After Di ff erence 685 935 545 700 480 770 475 640 680 800 685 955 590 780 600 790 630 830 Table 5: Volume (mL) of the right hind leg before and 8 hours after injection with a dose of rattlesnake venom and an antivenom. (e) The boxplot and normal quantile-quantile plot of the di ff erences are given in Figure 1 . Do these plots give any indication that the t procedures should not be used? 0 50 150 250 Change in Volume (ml) (a) The line represents the hypothesized value of 0. -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 150 200 250 Theoretical Quantiles Sample Quantiles (b) Normal quantile-quantile plot. Figure 1: Changes in volume (mL) of the right hind leg before and 8 hours after injection with a dose of rattlesnake venom and an antivenom. (f) Construct a 95% confidence interval for the true mean amount of swelling under the conditions of this experiment. Give an interpretation of the confidence interval in the context of the problem at hand. (g) Carry out a test of the null hypothesis that there is no swelling on average (after 8 hours), against the appropriate one-sided alternative hypothesis. Give the hypotheses in words and symbols, value of the test statistic, p -value, and conclusion. 10. Porter et al. ( 2010 ) investigated the e ff ect of a brief training program on the ability of health care professionals to detect deception. In part of the study, 26 health care workers were shown videos in which individuals sometimes showed a genuine smile, and sometimes showed a fake smile. Individuals completed this task before and after completing a 3 hour session designed to help them recognize deception. The participants’ responses were evaluated and they were given a discrimination accuracy score (high positive scores indicate a person is correctly discriminating between a genuine smile and a fake smile, values near 0 indicate the individual is not discriminating between a genuine smile and a fake smile, and negative values indicate that the individual is misclassifying genuine smiles as fake and fake smiles as genuine). The results of the study are given in Table 6 . The researchers were interested in testing whether the training program had an e ff ect, and in esti- 290 16s 190 190 200 Looksnormal as t.i.it p 2.2sse o Very small p very strong evidence again

Two Sample t-test t = -0.5692, df = 18, p-value = 0.5763 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.9817025 0.5631646 sample estimates: mean of x mean of y -0.12739020 0.08187872 Does this output yield strong evidence that ¯ X 1 does not equal ¯ X 2 ? 15. Suppose we wish to test the null hypothesis that μ 1 - μ 2 = 0 against a two-sided alternative hypoth- esis. (a) All else being equal, what will happen to the power of the test as the true di ff erence μ 1 - μ 2 gets closer to 0? (b) All else being equal, what will happen to the power of the test as the sample sizes increase? 16. Consider a two-sample t test of the null hypothesis of equal population means against a two-sided alternative. (a) Under what conditions would the test statistic be equal to 0? (b) Under what conditions would the p -value be equal to 1? (c) Under what conditions would the p -value be equal to 0? 17. Consider the three boxplots in Figure 2 , which represent samples of size 40 from 3 di ff erent popula- tions. Consider the following 3 null hypotheses (with two-sided alternatives in all cases). A B C -5 0 5 Figure 2 I. H 0 : μ A = μ B II. H 0 : μ A = μ C III. H 0 : μ B = μ C (a) Which test would result in the smallest p -value? (b) Which test would result in the largest p -value? 18. Consider the following output for a two-sample inference procedure, and the corresponding boxplots in Figure 3 . TPyt p Imy To to even lies close to a live it does w ̅ B II s as tw Ygi iit

A B 2 4 6 8 Figure 3 Two Sample t-test t = -1.0655, df = 43, p-value = 0.2926 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -1.6882973 0.5209971 sample estimates: mean of x mean of y 4.367041 4.950691 (a) Do the boxplots give any indication that this t procedure should not be used? (b) Is there strong evidence that the populations have di ff erent means? (c) Based on the confidence interval in the output, is there strong evidence that μ 1 - μ 2 6 = 10 ? (d) What is the sum of the two sample sizes? 19. Suppose we draw two independent samples of sizes n 1 = 100 and n 2 = 50 , and wish to test the null hypothesis that the population means are equal. The output for the two procedures (Welch and the pooled-variance t ) are: Output 1: data: sample1 and sample2 t = -1.3734, df = 98.659, p-value = 0.1727 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.5677384 0.1032975 sample estimates: mean of x mean of y -0.1298456 0.1023749 Output 2:

24. Test your conceptual understanding: Which of the following statements are true? You should be able to explain why a statement is true or why a statement is false. (a) If μ 1 = μ 2 , the sampling distribution of ¯ X 1 - ¯ X 2 is approximately symmetric about 0 for large sample sizes. (b) If ¯ X 1 = ¯ X 2 , the sampling distribution of μ 1 - μ 2 is approximately symmetric about 0 for large sample sizes. (c) SE ( ¯ X 1 - ¯ X 2 ) is the true standard deviation of the sampling distribution of ¯ X 1 - ¯ X 2 . (d) The pooled-variance t procedures work well, even when the variances for the two populations are very di ff erent, as long as the sample sizes are very di ff erent as well. (e) Suppose we are about to test the null hypothesis μ 1 = μ 2 against a two-sided alternative. All else being equal, the greater the di ff erence between μ 1 and μ 2 , the greater the power of the test. 25. Test your conceptual understanding: Which of the following statements are true? You should be able to explain why a statement is true or why a statement is false. (a) Suppose we are constructing a confidence interval for μ 1 - μ 2 . All else being equal, the greater the di ff erence between μ 1 and μ 2 , the wider the interval. (b) Suppose we are constructing a confidence interval for μ 1 - μ 2 . All else being equal, the greater the di ff erence between ¯ X 1 and ¯ X 2 , the wider the interval. (c) Suppose we are constructing a confidence interval for μ 1 - μ 2 . All else being equal, the greater the sample sizes, the narrower the interval. (d) Suppose we wish to test H 0 : μ 1 = μ 2 . We obtain random samples from the respective pop- ulations, run the appropriate test, and find that the p -value is 0.00000032. We can be very confident that our results have important practical implications. (e) If we test H 0 : μ 1 = μ 2 against a two-sided alternative and find a p -value of 0.32, then we know that μ 1 = μ 2 . 7.3 Applications 26. A study 2 investigated various aspects of tandem running in ants. Tandem running is a form of recruitment in which one ant with knowledge of the location of a food source or new nest site leads another ant to that location. (Optional: Watch an example of tandem running here: (1:53) ( http://www.youtube.com/watch?v=X2C7Sy2oPik )) Tandem running can be thought of as a form of teaching and learning. The study investigated various factors associated with tandem running, and investigated whether the age of the ant and the experience of the ant had any e ff ect on tandem running. Ants ( Temnothorax albipennis ) were categorized into 4 categories: Young and Inexperienced (YI), Young and Experienced (YE), Old and Inexperienced (OI), Old and Experienced (OE). One aspect of the study investigated the speed of tandem running when leading a YI ant. The mean speed of the run was recorded for 51 tandem runs in which the leader was YE, and 15 pairs of runs in which the leader was OE. The researchers were interested in a possible di ff erence in the mean speed of tandem running in these two situations. Figures 4 and 5 and Table 7 illustrate the data. (a) Do the plots give any indication that the t procedure should not be used? (b) Which version of the t procedure (pooled or unpooled) is more appropriate here? 2 Franklin et al. (2012). Do ants need to be old and experienced to teach? The Journal of Experimental Biology , 215:1287–1292 The data used here is simulated data based on their Figure 2, with similar results and conclusions. No roughly symmetric very diff SD so unpooled whelch is best

OE YE 0.5 1.0 1.5 2.0 2.5 Speed (mm/s) Figure 4: Speed of tandem running for ant pairs led by an old experienced (OE) ant, and ant paris led by a young experienced (YE) ant. -1 0 1 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Normal Q-Q Plot Theoretical Quantiles Sample Quantiles (a) OE ants. -2 -1 0 1 2 0.5 1.0 1.5 2.0 2.5 Normal Q-Q Plot Theoretical Quantiles Sample Quantiles (b) YE ants. Figure 5: Normal QQ plots for OE and YE ants. (c) Using the Welch procedure, construct a 95% confidence interval for the di ff erence in true mean tandem running speed between OE and YE ants. (Hint to ease the calculation burden: SE W ( ¯ X 1 - ¯ X 2 ) = 0 . 1272 , DF = 37 . 714 .) (d) The researchers were interested in investigating a possible di ff erence in the true mean run- ning speeds for the two types of ant. In words and symbols, what are the hypotheses of the appropriate hypothesis test? (e) What is the value of the appropriate test statistic? (f) What is the p -value of the test? Is there strong evidence against the null hypothesis? (g) The output from the statistical software R for the Welch procedure is: possible left skew 2.0249 0.1272 1.14 1.8 oasis www.as ff f o.m u ha Ha o.it 5189 an 2 1 1 5.18937.7141 7.487 10 6

Give a summary of the results of the analysis (including the results of the hypothesis test and the confidence interval). To what population do your conclusions apply? 28. Researchers used an experiment to investigate whether prolonged food restriction might lead to a relapse of drug abuse. 4 Rats were randomly assigned to a sated (well fed) group or a prolonged food restriction group (the rats were under food restriction for 10 days, and their weight fell to approximately 80% of the weight of the sated rats). Before the food deprivation period, all rats were trained to use a lever to self-administer heroin. Access to heroin was removed for all rats during the 10 day food deprivation period. Rats were then given access to the heroin lever, and the number of active lever presses in a 3 hour period was measured. The following table summarizes the results. Food restriction ¯ X 1 = 48 . 9 s 1 = 25 . 4 n 1 = 8 Sated (control) ¯ X 2 = 23 . 8 s 2 = 7 . 9 n 2 = 7 (a) What plots should be created before carrying out any statistical inference? (b) Plots of the data (not shown) show some right skewness in both groups, and so the use of the t procedures is a bit dubious. But if the skewness is similar in both groups, the t procedure may still perform reasonably well. Assuming that we choose to use the t procedures, which version of the t procedure (pooled or unpooled) is more appropriate here? (c) Using the Welch (unpooled variance) t procedure, construct a 95% confidence interval for the dif- ference in the means between the groups. (Hint to ease the calculation burden: SE W ( ¯ X 1 - ¯ X 2 ) = 9 . 4637 , DF = 8 . 512 .) (d) Suppose we wish to test whether there is strong evidence of a treatment e ff ect (a di ff erence between the true means of the two groups). In words and symbols, what are the hypotheses of the appropriate hypothesis test? (e) What is the value of the appropriate test statistic? (f) What is the p -value of the test? Is there strong evidence against the null hypothesis? (g) The output from the statistical software R for the pooled-variance procedure is: Welch Two Sample t-test data: Restricted and Sated t = 2.6523, df = 8.512, p-value = 0.02764 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 3.50315 46.69685 sample estimates: mean of x mean of y 48.9 23.8 Give a summary of the results of the analysis (including the results of the hypothesis test and the confidence interval). To what population do your conclusions apply? 29. A study 5 investigated several physical characteristics of the ears of Italian Caucasians. In one part of the study, 3D symmetry (a measure of the symmetry between the left and right ears) was measured on men and women between 31 and 40 years of age. Men ¯ X 1 = 95 . 71 s 1 = 1 . 50 n 1 = 66 Women ¯ X 2 = 95 . 41 s 2 = 1 . 64 n 2 = 28 4 Shalev (2011). Chronic food restriction augments the reinstatement of extinguished heroin-seeking behavior in rats. Addiction Biology , 17:691–693. 5 Sforza et al. (2009). Age- and sex-related changes in the normal human ear. Forensic Science International , 187:110.e1– 110.e7.

(a) What plots should be created before carrying out any inference procedures? (b) Which version of the t procedure (pooled or unpooled) is more appropriate here? (c) Using the pooled-variance t procedure, construct a 95% confidence interval for the di ff erence in the means of the 3D symmetry index. (Hint to ease the calculation burden: SE ( ¯ X 1 - ¯ X 2 ) = 0 . 3479 .) (d) Suppose we wish to test whether there is strong evidence of a di ff erence between the true mean 3D symmetry in men and women. In words and symbols, what are the hypotheses of the appropriate hypothesis test? (e) What is the value of the appropriate test statistic? (f) What is the p -value of the test? Is there strong evidence against the null hypothesis? (g) The output from the statistical software R for the pooled-variance procedure is: Two Sample t-test data: men and women t = 0.8624, df = 92, p-value = 0.3907 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.3908909 0.9908909 sample estimates: mean of x mean of y 95.71 95.41 Give a summary of the results of the analysis (including the results of the hypothesis test and the confidence interval). To what population do your conclusions apply? 30. A study 6 investigated several characteristics of psychopathic and nonpsychopathic male patients in a Dutch inpatient psychiatric treatment centre. In one part of the study, the BEST index (a measure of risk behaviours of psychiatric patients) was measured after several months of treatment. The higher the score on the BEST index, the better (less risky) the patient scored. The following table summarizes the results. Nonpsychopaths ¯ X 1 = 256 . 12 s 1 = 33 . 53 n 1 = 47 Psychopaths ¯ X 2 = 245 . 82 s 2 = 37 . 55 n 2 = 27 (a) What plots should be created before carrying out any inference procedures? (b) Which version of the t procedure (pooled or unpooled) is more appropriate here? (c) Using the pooled-variance t procedure, construct a 95% confidence interval for the di ff erence in the means of the BEST index. (Hint to ease the calculation burden: SE ( ¯ X 1 - ¯ X 2 ) = 8 . 4603 .) (d) Suppose we wish to test whether there is strong evidence of a di ff erence between the true means of the BEST index for psychopaths and nonpsychopaths. In words and symbols, what are the hypotheses of the appropriate hypothesis test? (e) What is the value of the appropriate test statistic? (f) What is the p -value of the test? Is there strong evidence against the null hypothesis? (g) The output from the statistical software R for the pooled-variance procedure is: Two Sample t-test data: psychopaths and non psychopaths t = 1.2174, df = 72, p-value = 0.2274 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -6.565314 27.165314 sample estimates: mean of x mean of y 256.12 245.82 6 Chakhssi et al. (2010). Change during forensic treatment in psychopathic versus nonpsychopathic o ff enders. 21:660–682. i I 8 803 1.217 2 1 pt 1.217,72 0.225

(b) Which one of the following is the most appropriate conclusion? i. There is very strong evidence that the population mean weight of babies born to mothers who used marijuana is equal to that of babies born to mothers who did not use marijuana. ii. There is not strong evidence of a di ff erence in population mean birth weight between the two groups. iii. There is strong evidence that the population mean birth weights for the two groups are equal. iv. There is not strong evidence that the sample mean birth weights are di ff erent. v. There is very strong evidence that marijuana causes a reduction in birth weight. 33. An experiment was designed to investigate the e ff ect of exercise on the size of tumours in rats. Thirty rats were injected with cancerous cells. These rats were then randomly assigned to two groups. Ten mice were kept in cages with exercise wheels, and 20 were kept in cages with no wheels. After six weeks, the diameter of the tumour (cm) was recorded. The following table summarizes the results: Exercise Wheels No Wheels Sample mean 1.2 1.5 Sample variance 0.08 0.12 (a) Is this an observational study or an experiment? (b) The researcher wanted to test whether the exercise wheels reduced the size of tumours in rats. Carry out the appropriate hypothesis test. Give appropriate hypotheses, the value of the test statistic, p -value, and an appropriate conclusion. (Assume equal population variances, and that the tumour sizes are normally distributed.) (Hint to ease the calculation burden: s 2 p = 0 . 10714 , SE ( ¯ X 1 - ¯ X 2 ) = 0 . 12677 ) (c) Suppose we found a p -value of .00000067 in the previous question. Would this give strong evidence of a causal link between exercise and size of tumour in this type of experiment? (d) Calculate a 90% confidence interval for the di ff erence in mean tumour size (for the pooled variance case). (e) Now, not assuming equal population variances, carry out the appropriate test. Give appropriate hypotheses, value of the test statistic, p -value, and an appropriate conclusion. (Hint to ease the calculation burden: DF = 21.764.) (f) Calculate a 90% confidence interval for the di ff erence in mean tumour size. (Not assuming equal population variances.) (Hint to ease the calculation burden: DF = 21.764.) (g) Which procedure (pooled-variance or Welch) is more appropriate in this case? 34. Researchers investigated the total cholesterol levels in the blood of male and female students at a large university. Total cholesterol (mg/dl) was measured on 26 male and 22 female student volunteers, with the following results. Males Females Sample mean 171.4 173.8 Sample standard deviation 32.9 34.1 Sample size 26 22 Although the people in the study are volunteers, for the purposes of these questions assume they can be thought of as random samples from the populations. (a) The researchers wanted to test the null hypothesis that the population mean total cholesterol level is the same for both males and females, against a two-sided alternative hypothesis. The following output represents the results of the pooled variance two-sample t procedure. experiment 20 Him u Ha I i test Dont havea butstill strong evidence wheel group have lower tumour men size ftp.t f ffm sqtl.o.es 287 1 t.ms nonlo.iaon iiiii iii Nota significant difference so up to preference I

Two Sample t-test data: males and females t = -0.2477, df = 46, p-value = 0.8055 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -21.90658 17.10658 sample estimates: mean of x mean of y 171.4 173.8 Of the following options, which one is the most appropriate conclusion at ↵ = . 05 ? i. There is strong evidence of a di ff erence in population means. ii. There is significant evidence that females have a higher population mean total cholesterol level. iii. The observed di ff erence in sample mean cholesterol levels is significant. iv. There is not significant evidence that the sample mean cholesterol levels are di ff erent. v. There is not significant evidence that the population mean cholesterol levels are di ff erent. (b) Give an appropriate interpretation of the confidence interval found in the output. 35. In an investigation into the e ff ectiveness of two processes that reduce contaminants in used motor oil, 18 batches of used motor oil were randomly assigned to the two processes. 12 were randomly assigned to process A, and 6 to process B. The contaminant level after processing was measured for both processes. The results are given in Table 9 . A B Sample size 12 6 Sample variance 16 9 Sample mean 63 57 Table 9: Summary of information for the motor oil question. Suppose that the normality assumption of the t procedures is in fact reasonable, and that the population variance of the contaminants is the same for both processing methods. (a) Calculate a 95% confidence interval for the di ff erence in the population mean contaminant levels ( μ A - μ B ). Give an appropriate interpretation of the interval. (b) Carry out a test of the null hypothesis that the true mean contaminant level is the same for both processes, against the alternative hypothesis that it is di ff erent. Give appropriate hypotheses in words and symbols, value of the test statistic, and p -value. (c) Summarize the results of the analysis. 36. Many studies have investigated a connection between “fear of negative evaluation” and bulimia. Suppose that researchers at a large university are interested in carrying out their own investigation. Eleven female students with bulimia completed a questionnaire and were assigned a “fear of negative evaluation” score, with a resulting sample mean of 19.7. Fourteen female students with normal eating habits were given the same questionnaire, with a resulting sample mean “fear of negative evaluation” score of 14.9. The researchers ran a pooled-variance two-sample t procedure on the data, with the following results.

(b) As part of their analysis into the di ff erences between the resistors, the researchers want to test the null hypothesis that the population mean resistance is the same for both types of resistor, against the alternative hypothesis that the mean resistance is di ff erent. They also wish to calculate a 95% confidence interval for the di ff erence in the mean resistance. They decide to use the pooled-variance t procedure . The Welch procedure would have also been a reasonable choice (the output for the Welch procedure is included farther below). What is the value of the pooled sample variance ( s 2 p )? (c) What are the appropriate hypotheses in words and symbols? (d) What is the value of the appropriate t statistic? (e) Give an appropriate conclusion at the 5% significance level. (f) What is a 95% confidence interval calculated using the pooled-variance t procedure? (g) The following output represents the results of the Welch procedure on the data above, using the alternative hypothesis that the population mean resistance for the two types of resistors is di ff erent. Welch Two Sample t-test data: resistors t = -2.7904, df = 17.695, p-value = 0.01222 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -1.0522974 -0.1477026 sample estimates: mean of x mean of y 11.6 12.2 Based on the confidence interval in the output, what can be said about possible di ff erences in the mean resistance? (h) Refer again to the output from the Welch procedure. Note that the output gives the p -value for a two-sided alternative. Had the researchers felt that H a : μ A < μ B was the more appropriate alternative (before looking at the data), what would the p -value of that test be? 38. A company is interested in investigating properties of a new aluminum alloy that is produced by a new experimental process. They wish to compare the yield strength of this new alloy to the yield strength of a standard alloy that is commonly used. The following table gives a summary of the yield strength (MPa) for independent samples of each type of alloy. Alloy type Sample size Sample mean Sample standard deviation New alloy 80 664.2 23.3 Standard alloy 75 624.3 19.8 Suppose it is reasonable to assume that the alloy strengths are approximately normally distributed, and we wish to carry out inference procedures to investigate possible di ff erences between the alloys. There is little di ff erence between the sample standard deviations, so the use of the pooled-variance t procedure is reasonable here. Use the pooled-variance t procedure to answer the following questions. (a) Calculate a 95% confidence interval for the di ff erence in true mean yield strength. Give a proper interpretation of the interval. (Hint to ease the calculation burden: s 2 p = 469 . 9299 , SE ( ¯ X 1 - ¯ X 2 ) = 3 . 4842 .) (b) Carry out a test of the null hypothesis that the population means are equal, against a two-sided alternative hypothesis. Give appropriate hypotheses (in words and symbols), test statistic, p -value, and conclusion.

39. A person suspects that a certain grocery store is overstating the weight of fresh chickens, and are thus overcharging their customers. He randomly samples five chicken packages, records the weight stated on the package (in grams) and the true weight of the chicken. The results are illustrated in Table 10 . Chicken Stated weight Actual weight 1 908 866 2 1120 1087 3 795 783 4 912 890 5 1402 1397 Table 10: Stated weight and actual weight for a sample of five chickens. Even though it is somewhat dubious to use the t procedures for such a small sample size, the person conducting the investigation feels the normality assumption is reasonable, and goes ahead with the t procedures. He find the output: Paired t-test data: stated and actual t = 3.3861, df = 4, p-value = 0.02763 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 4.104811 41.495189 sample estimates: mean of the differences 22.8 (a) Give a summary of the results of the analysis. (b) If the normality assumption is not in fact reasonable, what are the consequences? 40. Researchers are investigating whether a new experimental method of measuring viscosity in gases is as e ff ective as an older method. They take measurements of viscosity on 6 samples of gas, using both the new method and the older method. The results are given in Table 11 . The researchers want to test whether there is a di ff erence between the two methods, and estimate the size of the di ff erence with a confidence interval. Gas Sample Old Method New Method 1 2.74 2.71 2 3.11 3.01 3 2.99 3.11 4 2.99 2.94 5 2.88 2.96 6 3.29 3.21 Table 11: Measures of viscosity in gases. (a) Explain why a paired-di ff erence procedure should be used in this situation. (b) If we want to use a t inference procedure, how could we assess if the normality assumption is reasonable? For the following questions, suppose that the normality assumption is reasonable.