Module 9 Exam Spring 2021

1 Spring 2021 Module 9 Exam Statement of Academic Honesty When I submit my answers to this exam, I am declaring that my answers are due to my work, and my work alone. I did not ask for help on any question or problem on this exam from anyone else, including any source (person or other resource) from the internet. I did not request someone else share their work or their answers with me, and I did not share my answers with anyone else. I do understand that I can ask my instructor with my questions about problems on the exam. Name: _____________________________________________

2 Spring 2021 Confidence Interval and Hypothesis Testing for the Population Mean, µ: A confidence interval for the population mean, µ, has the form: �𝑿𝑿 � − 𝒕𝒕 𝒄𝒄 𝑺𝑺 √𝒏𝒏 , 𝑿𝑿 � + 𝒕𝒕 𝒄𝒄 𝑺𝑺 √𝒏𝒏 � The critical t-value T c is found from the t-distribution table with degrees of freedom equal to n-1 , or by using invT() on the TI calculator, or by using StatKey ( http://www.lock5stat.com/StatKey/ ). The margin of error is given by 𝑬𝑬 = 𝑻𝑻 𝒄𝒄 𝑺𝑺 √𝒏𝒏 . You may check your work with TI: STAT, TESTS; TInterval For Hypothesis Testing the test-statistic is given by 𝒕𝒕 = 𝑿𝑿 � −𝝁𝝁 𝒔𝒔 √𝒏𝒏 � in which µ is the assumed value of the population mean in the null hypothesis. The p-value is calculated by using tcdf(Low,High,df) . Note: for a two-tailed test you need to multiply this value by 2. You may check your work with TI: STAT, TESTS; T-Test Confidence Interval and Hypothesis Testing for the Difference Between Two Population Means, µ 1 -µ 2 : A confidence interval for the difference between two population means, µ 1 -µ 2 , has the form: � ( 𝒙𝒙 � 𝟏𝟏 − 𝒙𝒙 � 𝟐𝟐 ) − 𝑬𝑬 , ( 𝒙𝒙 � 𝟏𝟏 − 𝒙𝒙 � 𝟐𝟐 ) + 𝑬𝑬� where 𝑬𝑬 = 𝑻𝑻 𝑪𝑪 � 𝒔𝒔 𝟏𝟏 𝟐𝟐 𝒏𝒏 𝟏𝟏 + 𝒔𝒔 𝟐𝟐 𝟐𝟐 𝒏𝒏 𝟐𝟐 and the critical t-value, T c , is found from the t-distribution table, or by using invT() on the TI calculator, or by using StatKey ( http://www.lock5stat.com/StatKey/ ). The degrees of freedom value will be given in the problem (since it is messy to calculate). You may check your work with TI: STAT, TESTS; 2-SampTInt For Hypothesis Testing the test-statistic is given by 𝒕𝒕 = ( 𝒙𝒙 � 𝟏𝟏 −𝒙𝒙 � 𝟐𝟐 ) − ( 𝝁𝝁 𝟏𝟏 −𝝁𝝁 𝟐𝟐 ) � 𝒔𝒔 𝟏𝟏 𝟐𝟐 𝒏𝒏 𝟏𝟏 + 𝒔𝒔 𝟐𝟐 𝟐𝟐 𝒏𝒏 𝟐𝟐 in which µ 1 -µ 2 is the assumed value of the difference in population means, which is always 0 in our problems. The p-value is calculated by using tcdf(Low,High,df) . Note: for a two-tailed test you need to multiply this value by 2. You may check your work with TI: STAT, TESTS; 2-SampTTest

4 Spring 2021 1. In a soft drink bottling plant, the machinery should fill the cans with 355 milliliters of soda. A consumer group has complained that the cans are being under-filled (have less than 355 ml). a. State the null hypothesis and the alternative hypothesis that should be used to determine if the soda cans are being correctly filled or underfilled (less than 355 ml, on average). (25 pts) H 0 : English version: Symbolic version: H 1 : English version: Symbolic version: A random sample of 22 cans are measured and the average soda in the cans was 353 milliliters (ml) with a standard deviation of 4 milliliters. b. The sample size is rather small. What assumption is needed so that the normality criteria are satisfied? c. Calculate the value of the test-statistic, t, using 𝒕𝒕 = 𝒙𝒙 �−𝝁𝝁 𝒔𝒔 √𝒏𝒏 � . d. Use the test-statistic and tcdf(Low,High,df) to find the p-value for this sample.

5 Spring 2021 e. At a significance level of 0.05, should we reject or keep the null hypothesis? f. Explain your conclusion in words a non-statistics person would understand. 2. In a study published in the Journal of the American Medical Association in 2007, the weight loss of women on various diets was reported. The following table gives the results for women on a low-carbohydrate diet and on a low-fat diet. The weight loss values are for a 6- month period of being on the diet plan: Diet Group n Average weight loss Standard Deviation Low Carbohydrate 77 4.7 kg 7.2 kg Low Fat 79 2.6 kg 5.9 kg (20 pts) a. For this problem, the degrees of freedom is 147, which gives the critical t-value for a 95% confidence interval to be T C = 1.992 . Use this value and the values in the table to calculate the margin of error for the difference in population means: 𝑬𝑬 = 𝑻𝑻 𝑪𝑪 � 𝒔𝒔 𝟏𝟏 𝟐𝟐 𝒏𝒏 𝟏𝟏 + 𝒔𝒔 𝟐𝟐 𝟐𝟐 𝒏𝒏 𝟐𝟐 . b. Calculate the difference in sample means, 𝒙𝒙 � 𝟏𝟏 − 𝒙𝒙 � 𝟐𝟐 , using the low-carbohydrate group as group 1 and the low-fat group as group 2. Then calculate the 95% confidence interval for the difference in population means, 𝝁𝝁 𝟏𝟏 − 𝝁𝝁 𝟐𝟐 .

7 Spring 2021 3. A Nursing instructor asks her students to report measurements of body fat percentages that they had performed on randomly chosen adults (not necessarily the students). The following 16 values of body fat percentages represent a random sample out of that group. 22.4 20.1 34.5 26.8 18.2 23.6 26.1 21.3 13.6 20.8 22.1 32.8 17.0 11.0 5.2 31.9 (25 pts) a. What do we need to assume so that the normality criteria are met? b. Calculate the sample statistics, 𝑥𝑥̅ , s and n . c. Calculate, or use the table of critical t-values at the front of the exam to find, the critical t value, T C , for a 90% level of confidence and the degrees of freedom for the sample. d. Calculate the margin of error, E, for a 90% confidence interval. e. Construct a 90% confidence interval for the body fat percentages for adults. f. Explain what your confidence interval means in words a non-statistics person would understand.

8 Spring 2021 4. Honda Civics and Toyota Corollas are two car models that are often considered very similar to each other. In a comparison of fuel economy, the miles per gallon (mpg) used by these cars was measured. A random sample of 38 different Honda Civics of various types and 14 different Toyota Corollas of various types were chosen and their gasoline mileage measured under controlled conditions. The following is a summary of the results. Sample Average Sample Standard Deviation Sample Size Toyota Corollas 31.6 mpg 1.4 mpg 14 Honda Civics 33.7 mpg 1.4 mpg 38 (30 pts) a. Write the null and alternative hypothesis for deciding: is the population mean miles per gallon of Toyota Corollas lower than that of Honda Civics. H 0 : English: Symbolic: H 1 : English: Symbolic: b. Calculate the difference of sample means, 𝒙𝒙 � 𝑻𝑻 − 𝒙𝒙 � 𝑯𝑯 , and calculate the standard error 𝝈𝝈 = � 𝒔𝒔 𝑻𝑻 𝟐𝟐 𝒏𝒏 𝑻𝑻 + 𝒔𝒔 𝑯𝑯 𝟐𝟐 𝒏𝒏 𝑯𝑯 . c. Calculate the t-score for your difference from part b: 𝒕𝒕 = ( 𝒙𝒙 � 𝑻𝑻 −𝒙𝒙 � 𝑯𝑯 ) − ( 𝝁𝝁 𝑻𝑻 −𝝁𝝁 𝑯𝑯 ) 𝝈𝝈 .