SHOW ALL WORK, ENTER ANSWERS ON THE BLANK PROVIDED Name SB ID #:_ _ Your TA This is worth 5% of your final exam grade.

SHOW ALL WORK, ENTER ANSWERS ON THE BLANK PROVIDED Name SB ID #:_ _ Your TA This is worth 5% of your final exam grade. Please print and hand in this original document with your work on it. Show all work in any blank space provided on each page and enter your final answer on the blank line provided. You may check your work with a calculator, but all computations must be shown. You are expected to work independently, not with the TA 1. (14 points) A school administrator wants to compare two reading methods: Method S and method B. A class of 44 students is randomly divided into two groups of 22 each. One is taught with method S and the other is taught with B. Is the mean reading score with S higher than with B? The sample mean reading score for method S was 43.0691, with a standard deviation of 5.5641 and the sample mean reading score for method B was 40.0768 with a standard deviation of 5.2247. These are sample standard deviations. No alpha is provided for this problem. State the null and the alternate hypotheses Ho: versus Ha: ____________ Assume that the data are independent samples from normal populations with equal population variances. Thus, we perform a pooled variance t test. Find an estimate of the pooled variance. ______________ (4 decimal places) Find sp (3 decimals) ____________________ Degrees of Freedom = ______________ The critical value(s) from the t table at the appropriate degrees of freedom are given by ____________________ The t test statistic is ______________ Estimate the p-value______________ State and interpret your decision: ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ Previously, you were asked to perform a pooled variance t-test. Were we justified in assuming equal variances? To answer this question please conduct a hypothesis test (F test) to check for equality of variances. State the null and alternate hypotheses Ho: ___________________versus Ha: ______________________ State the F test statistic _________________________ The numerator and denominator degrees of freedom______,_____ State the critical value obtained from the F table ____________ (F table is posted under Lectures on Blackboard for your convenience.) State your conclusion of this F test. Were we justified in conducting a pooled variance t test? _______________________________________________________ ___________________________________________________(Yes/NO) If the pooled variance t test is not warranted, (i.e. if variances are not the same), what test is one supposed to use to compare the means? ______________________________________________________________ TOTAL POINTS FOR #1________ 2. (11 points) David wants to compute whether an alternate route to work would be quicker. For the next 58 working days he randomly divides his observations into two groups of 29 days each. He drives the first route for one set of 29 days and drives the second for the other set of 29 days recording, for each day, the commuting times in minutes. Assume that times of commute are normal. Route 1 n=29 sample mean = 31.945 s=5.87 Route 2 n=29 sample mean = 28.105 s=6.24 a. Does the assumption of common population standard deviation seem plausible? State the null and alternate hypotheses Ho: versus Ha:_____________ b. The numerator and denominator degrees of freedom______,______ c. Give the test statistic_________________ d. What is your decision regarding common variance ___________________________________________________________________ ___________________________________________________________________ e. State the hypothesis to test if route two is quicker than route one. Ho: Ha:_________________________ f. The test statistic is ____ g. The p-value is h. The degrees of freedom are_______________ i. The decision is __________________________________________________________ Route 2 (does/does not) appear to be quicker. j. State the F critical value from the F table that was used in part d____________ k. State the pooled variance__________________ TOTAL POINTS FOR #2________ 3. (13 points) We would like to compare the characteristics of tuberculosis meningitis in patients infected with HIV and those who are not infected. In particular, we would like to determine whether the two populations have the same mean age. A sample of 37 infected patients has mean age 27.9 years and a standard deviation 5.6 years. A sample of 19 patients who are not infected has mean age 38.8 years and standard deviation 21.7 years. Alpha = .05 Conduct a hypothesis test to check for equality of variances. State the null and alternate hypotheses Ho___________________ versus Ha:__________ Compute the F test statistic ___________ State the numerator and denominator deg of freedom______,_____ State the critical value obtained from the F table ____________ State your conclusion regarding the variances. _______________________________________________________ _______________________________________________________ _______________________________________________________ Based on (a), what test will you use to determine if the 2 populations have the same mean age? _______________________________________________________ We continue with the test on the next page: State Ho: ______________________ versus Ha: ______________________ Find the test statistic __________________ State the degrees of freedom (Welch/Satterthwaite)_____________ State the critical value(s) _____________ State your conclusion, and justify your conclusion using the critical values and the test statistic. _______________________________________________________ _______________________________________________________ _______________________________________________________ Approximate the P-value for this test. ______________________ State the confidence interval for the difference in means________,________ TOTAL POINTS FOR #3________ 4. (18 points) Company officials were concerned about the length of time a particular drug product retained its potency. A random sample of n1 = 10 bottles of the product was drawn from the production line and analyzed for potency. A second sample of n2 = 10 bottles was obtained and stored in a regulated environment for 1 year. Let 1 denote the mean potency for all bottles coming off the production line and let 2 denote the mean potency for all bottles retained for a period of 1 year. Assume that population variances are unknown but equal. Is the potency for fresh drugs different from that for stored drugs? Use alpha = .05 Potency Reading for the two samples: Fresh: 10.2 10.5 10.3 10.8 9.8 10.6 10.7 10.2 10 10.6 Stored: 9.8 9.7 9.6 9.5 10.1 9.6 10.2 9.8 10.1 9.9 Ho = ____________ Ha = ____________ Find the sample mean for fresh bottles____________ ( 2 decimal places) Find the sample mean for stored bottles__________ ( 2 decimal places) Find the sample standard deviation, s1 for fresh bottles_________ ( 4 decimal places) Find the sample standard deviation, s2 for stored bottles________ ( 4 decimal places) Compute the 95% confidence interval for 1 - 2 _____________ Find an estimate of the pooled variance. ______________ (4 decimal places) Find sp (3 decimals) ____________________ Degrees of Freedom = ______________ the critical values from the t table at the appropriate degrees of freedom are given by ____________________ The t test statistic is ______________ Estimate the p-value______________ State and interpret your decision: ______________________________________________ In this problem you were asked to perform a pooled variance t-test. Were we justified in assuming equal variances? To answer this question, conduct a hypothesis test (F test) to check for equality of variances. State the null and alternate hypotheses Ho: Ha: ________ State the F test statistic _________________________ The numerator and denominator deg of freedom , ____ State the critical value obtained from the F table ____________ (F table is posted under Lectures on Blackboard for your convenience.) State your conclusion of this F test. Were we justified in conducting a pooled variance t test? _______________________________________________________ _______________________________________________________ _______________________________________________________ ______________________________________________(Yes/NO) TOTAL POINTS FOR #4________ 5. (9 points) Consider the paired data shown here: Pair X Y Difference 1 21 29 2 28 30 3 17 21 4 24 25 5 27 33 In order to test whether the means are equal or not, a t test is performed at alpha = 1%. a. Ho: versus Ha: ____________ b. The mean of the differences = ___________ c. Standard deviation of the differences = _________ (4 decimal places) d. Degrees of freedom = ____________ e. t test statistic = _____________ f. Critical value(s) for this test = ____________ g. Decision: ____________________________ h. Estimate the P-value_______________ i. Appropriate confidence interval for the difference in means: (________, _______) TOTAL POINTS FOR #5________ 6. (7 points) From a socio-demographic Risk factors in spousal abuse study, it is observed that the distribution of ages in a sample is as follows: Age Number in sample 20-24 103 25-34 216 35-44 171 Total Conduct a chi-square goodness of fit at the 10 % significance level to test whether the sample provides a good match to the known age distribution in the population. The known percentages are 18%, 50%, 32% Ho: versus Ha: ____________ The expected number in each category is _, , _______ Which group above seems to be underrepresented in the sample?_____________ The chi-squared test statistic is ________________ The critical value from the chi-squared table is _____________ The p-value is ___________ Will you reject Ho? Explain in the context of the problem _______________________________________________________________ ____________________________________________________________________________ TOTAL POINTS FOR #6________ 7. (8 points) Sample data has been collected in a study of calcium supplements and their effects on blood pressure. A placebo group and a calcium group began the study with blood pressure measurements. The sample values are listed below. At the .05 significance level, test the claim that the two sample groups come from populations with the same standard deviation. Placebo: 124.6 104.8 96.5 116.3 106.1 128.8 107.2 123.1 118.1 108.5 120.4 122.5 113.6 Calcium: 129.1 123.4 102.7 118.1 114.7 120.9 104.4 116.3 109.6 127.7 108 124.3 106.6 121.4 113.2 Find the sample standard deviation, for Placebo_________ ( 4 decimal places) Find the sample standard deviation, for Calcium_________ ( 4 decimal places) State the null and alternate hypotheses Ho: Ha: ________ Find the test statistic Find the critical value State the appropriate degrees of freedom Draw the appropriate distribution showing the proper critical value. Label all necessary components. What is your conclusion? Explain TOTAL POINTS FOR #7________ THIS PAPER WAS GRADED BY __________________________

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