Stats 250 Practice Concepts Exam 2 F23

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 3 of 22 4. Waste Watch - Waste Watch is a non-profit organization that promotes environmentally friendly methods of waste management. To find out whether or not there is a relationship between waste treatment preferences and city (Smogville versus Litterview), Waste Watch obtained a sample of residents from these two cities. Along with the city of residence, each resident was classified according to their primary method for treating waste. The data and output are as follows: Primary method for treating waste City Incinerator Landfill Compost/Recycle Total Smogville 38 41 21 100 Litterview 28 41 31 100 Total 66 82 52 200 Using the information provided, you are asked to assess if there is a relationship between waste treatment preferences and city of residence for the population of residents represented by the sample. 1) State the appropriate null hypothesis . ] 2) You may assume you have a random sample of results. Check the remaining necessary assumption(s) (show all work) . 3) Perform the appropriate test by providing the test statistic value and the corresponding p -value (show all work). 4) Evaluate the p-value and provide a conclusion in context. You have this page and the next page to provide your answer which needs to be well organized and clearly use labels for your Steps 1, 2, 3, and 4.

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 6 of 22 Expected test statistic value: _______________________________ b. Provide a conclusion. 8. Gas Mileage - Each year, the U.S. Department of Energy publishes fuel economy information for new model cars. The data for 2017 car models were released. A random sample of 51 4-cylinder cars was selected, and the city fuel economy (in miles per gallon (MPG)) and the highway fuel economy (in miles per gallon (MPG)) were recorded. A scatterplot of highway MPG versus city MPG is provided along with some R output. a. From the scatterplot, we observe a moderately strong linear relationship. What are the other two observations that should be made from this plot in this initial stage of the regression analysis? 1:_____________________________________________________ 2: _____________________________________________________ b. Give the value of the correlation coefficient between city fuel economy and highway fuel economy. Then, select the appropriate units for this value. (place a clear X on the appropriate line) Value: _________________ c. One car in the sample gets 25 city MPG and 33 highway MPG. Provide the predicted highway MPG for such a car and calculate the residual for this car. Include units . Predicted value: ______________________________ Residual: __________________________________ Scatterplot of fuel economy by MB Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.70536 2.36724 -0.298 0.767 ** City.MPG 1.40031 0.09985 14.025 <2.2e-16 ** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 2.185 on 49 degrees of freedom Multiple R-squared: 0.8006, Adjusted R-squared: 0.7965 F-statistic: 196.7 on 1 and 49 DF, p-value: <2.2e-16 Highway fuel economy (in miles per gallon) City fuel economy (in miles per gallon)

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 9 of 22 e. Provide a conclusion. 10. Lunch Drink— Is there a relationship between lunch drink preference (Coffee, Juice, Soft Drink, Tea, or Water) and age when considering three different age groups (child, teenagers, and adults) for all residents in Michigan? To assess if there is an association between lunch drink preference and age groups for all residents of Michigan, we will be conducting a chi-square test of independence. a. Assuming that there is no association between lunch drink preference and age groups for all residents of Michigan, what is the expected test statistic value ? Final Answer: __________________________________ b. Assuming that there is no association between lunch drink preference and age groups for all residents of Michigan, what would be the median value of the test statistic? ____ 0 ____ 7.244 ____ 8 ___ 10.219 c. Assuming that there is no association between lunch drink preference and age groups for all residents of Michigan, what would be value of the 75 th percentile for the test statistic? ____ 0 ____ 7.244 ____ 8 ___ 10.219 d. State all the assumptions that must be met in order to conduct the chi-square test of independence. e. The resulting test statistic value for this chi-square test of homogeneity is 15.69 and the corresponding p - value is 0.047. Provide an interpretation of the p -value in context.

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 12 of 22 b. Supple repeated samples of 30 tent stakes are selected at random and the sample mean length is computed for each sample of 30 tent stakes. What is the approximate probability that a mean length for a sample of 30 tent stakes is more than 43.5 inches? Final Answer = ____________________ 15. Pet Preference and Age - A researcher was interested in whether there is a relationship between a person’s age and their desire to have a pet. A random sample of 500 Michigan adults was selected and two variables were recorded, age group and pet preference. Age group was defined as older adults (60 years of age or older) versus younger adults (18 years of age up to but not including 60 years of age). Pet preference was defined as the responses to the question “If you could own only one pet, what kind would you choose?” The possible options were “dog,” “cat,” “other pet,” and “no pet.” The table provides the results a. Based on these results, if we randomly selected an older adult, what is the probability that this older adult will respond as preferring to own a dog? b. Based on these results, what is the probability of the event “preferring to own a cat” or “person is a younger adult”? c. What would be the appropriate Chi-square test for assessing if there is a relationship between a person’s age group and their pet preference? _____ Goodness-of-fit _____ Test of Independence d. Assuming there is no relationship between a person’s age group and their pet preference, how many older adults would we have expected to say “dog”?

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 15 of 22 17. Living arrangements We are interested in understanding the relationship between the highest degree received and living arrangements among all adults in the US. We take a random sample of 300 adults, and from each subject, we record the highest degree received (High School, Bachelor’s, Master’s, PhD) and living arrangements (own, rent from a private proprietor, rent-free). The results are provided below: Living Arrangements Highest degree received High School Bachelors Masters PhD Total Own 35 24 27 14 100 Rent from a private proprietor 17 42 27 14 100 Rent-free 26 33 27 14 100 Total 78 99 81 42 300 Using this data, we would like to assess if there is a relationship between the highest degree earned and living arrangements among all adults in the US. 1) State the appropriate null hypothesis in context. ] 2) You may assume you have a random sample of results. Check the remaining necessary assumption(s) (show all work) . 3) Perform the appropriate test by providing the test statistic value and the corresponding p -value (show all work). 4) Evaluate the p-value and provide a conclusion . You have this page and the next page to provide your answer which needs to be well organized and clearly use labels for your Steps 1, 2, 3, and 4.

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 18 of 22 20. Travel Agency Name that Scenario - One important aspect in statistics is to understand which statistical methods or procedures are appropriate to use to address the research problem or question of interest. For each description of a research question below, clearly select and write the letter corresponding to the statistical analysis technique most appropriate for addressing that research question. ____ 1. In a recent study 45 randomly selected travel agency customers of were asked to review a new feature on the agency website for finding vacation deals. The results will be used to assess if more than 80% of all customers find the new feature easy to navigate. A. 1-sample t-test for a population mean B. Paired t-test for a population mean difference C. 2-sample t-test for the comparison of two population means D. Regression E. 1-sample Z-test for a population proportion F. 2-sample Z-test for the comparison of two population proportions G. Chi-squared test of independence H. Chi-squared test of goodness of fit ____ 2. A travel agent wants to assess if the preferred cruise line (Norwegian, Holland, Carnival, Royal Caribbean, Other) is associated with the age group of the customer (young adult, middle-aged adult, older adult). ____ 3. The agency has created two new vacation package options (Option 1 and Option 2) for Caribbean Cruise. A random sample of 100 customers planning a Caribbean Cruise was selected, and for each customer we recorded the vacation package option selected. The travel agent would like to assess if a majority of customers prefer Option 1 over Option 2 for all Caribbean Cruise customers. ____ 4. A survey of 100 families who booked a vacation at a Disney World Resort is conducted to assess if the average length of stay is more than 4 days. ____ 5. A student intern at the travel agency designed an experiment to assess if the waiting time at the downtown location is less than that at the satellite location, on average. She plans to collect a random sample of 40 customers who visit the downtown location and an independent random sample of 40 customers who visit the satellite location and record for each the number of minutes the customer had to wait before being helped by an agent. ____ 6. A travel agent wants to assess if there is a significant difference in the average length of stay across these 2 different types of accommodations (major hotel chains and locally owned bed and breakfast) for all tourists visiting coastal cities in Florida?

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 21 of 22 c. Which among them most confirms (or replicates) the findings of the study results in (b)? Provide a 2-3 sentence justification of your choice that compares test statistics, p-values, and estimated effect sizes across the studies. d. Which of the three follow-up studies do you expect was based on the largest overall sample size? Provide a 2- 3 sentence justification of your choice. 23. Sleep Apnea ~ Researchers recently investigated 1 sleep-related breathing disorders in adults with traumatic brain injuries (TBI). The respiratory disturbance index (RDI) represents the number of apneic and hypopneic episodes (snoring, gurgling, air passage obstruction, etc.) per hour of sleep and was used by scientists to measure the severity of a patient’s sleep apnea. The recorded measures for male and female patients are offered in the table above. a. Suppose the researchers want to test to see if there is a meaningful difference in mean RDI levels between male and female patients with traumatic brain injuries. What are the appropriate hypotheses to test? ࠵? ! : __________________________ vs ࠵? " : _________________________ b. The test of the correct hypotheses in (a) yield a t-test statistic of ࠵? = −2.2923 with ࠵? = 0.038 and ࠵? " = −0.8658 , suggesting that there is strong evidence of a difference of mean RDI levels across male and female patients with traumatic brain injuries, and this the effect size describing the difference between these means is quite large. Suppose that a second set of researchers hopes to replicate the findings of the initial study. The tables below show three hypothetical results the researchers might collect, where result A differs in sample size, result B differs in 1 “Sleep Apnea in Adults With Traumatic Brain Injury: A Preliminary Investigation.” Archives of Physical Medicine and Rehabilitation , Vol. 82(3), 316-321. Group Mean ࠵? 8 Standard deviation ࠵? Sample size ࠵? Female 0.9 0.98 7 Male 1.97 1.30 21

Stats 250: practice problems to reinforce exam 2 material—Fall 23 Page 22 of 22 estimated data variability, and result C differs in difference in mean groups outcomes. The initial study results are reprinted for easy comparison. No computations are required for the exercises below. Initial ࠵?̅ ࠵? ࠵? A ࠵?̅ ࠵? ࠵? B ࠵?̅ ࠵? ࠵? C ࠵?̅ ࠵? ࠵? ࠵? 0.9 0.98 7 ࠵? 0.9 0.98 70 ࠵? 0.9 2.98 7 ࠵? 0.5 0.98 7 ࠵? 1.97 1.30 21 ࠵? 1.97 1.30 210 ࠵? 1.97 3.30 21 ࠵? 2.97 1.30 21 i. Select the study that most replicates the findings of the initial study on sleep apnea. Select only one. o Study A o Study B o Study C ii. Without doing any computations, which of the replication studies would produce an est. effect size ࠵? " that is identical to the one from the initial study? Select only one. o Study A o Study B o Study C iii. Which of the replication studies would produce a p-value that is larger than the one from the initial study? Select all that apply. o Study A o Study B o Study C iv. Which of the replication studies would produce a p-value that is smaller than the one from the initial study? Select all that apply. o Study A o Study B o Study C