4.A researcher wanted to study the effect of a newly developed gasoline additive (Additive X) on automobile mileage (miles per gallon, MPG). To gather information, a random sample of cars has been selected and their gasoline mileages (MPG) have been determined. For each car, the MPG was measured both when gasoline without Additive X is used and when gasoline with Additive X is used. The order of the two treatments(with Additive X versus without Additive X) was randomized and care was taken so that there was no carry over effect between the two measurements. The 90% confidence interval for the average improvement in mileage (difference = MPG with Additive XlessMPG without Additive X) for the population of all such cars was found to be (1.5 MPG, 3.5 MPG).

The margin of error of this interval is:

• 1
• 1.5
• 2
• 3.5
• 90
• Cannot be determined since we do not know the sample size n.

Acompetitorsfuel additivehad an established average improvement in mileage of 1 MPG. The researcher would like to use the confidence interval to test the hypotheses H₀:= 1 MPG versus the alternative H_a: 1 MPG, at a 10% level of significance. Here the parameterrepresents the mean improvement in mileage (with Additive X versus without Additive X) for the population of all such cars. What would be her decision?

• reject the null hypothesis
• fail to reject the null hypothesis
• cannotbedeterminedsincethesamplesizeisnotknown
• Youinterview100UniversityofMichiganstudentsintheirfreshmanyearandagainintheirsenioryearandaskeachaboutthetypicalnumberofminutesperdaythattheyspendusingsocialmedia.Let_drepresentthepopulationmeandifferenceintimespentusingsocialmedia(freshman-senior).YouwishtotestthehypothesesH₀:_d=0versusH_a:_d0.Youcomputethe95%confidenceintervalfor_dandfindittobe(13.2minutes,24.8minutes).Assumethenecessaryconditionsforconductingtheconfidenceintervalandtestaremet.Whichofthefollowingarecorrectstatements?Selectallthatarecorrect.
• You would reject the null hypothesis at the 5% level.
• Your would fail to reject the null hypothesis at the 5% level.
• The p-value for the test is more than 0.05.
• A mistake has been made as the confidence interval should contain 0 about 95% of the time.

For quantitative data with two sets of observations, consider the two following approaches for analysis.

(1) The two sets of quantitative observations areindependentand viewed as independent random samples selected from two separate populations in order to estimate₁-₂with a confidence interval or test hypotheses about how the two population mean responses compare.

(2) The two sets of quantitative observations are dependent and viewed as coming from one population ofpairedobservations.

The focus is on the difference between the two observations (pairwise) and the researcher is interested in estimating the population mean difference_dor testing hypotheses about the population mean difference. Consider the following scenarios and decide whether each reflects an analysis that should be focused on thedifference in two population means (₁-₂)or thepopulation mean difference (_d).

8.a 1 point(s)

A person is considered hypertensive if their diastolic blood pressure is elevated above 90 mmHg. A random sample of 30 hypertensive patients is selected and their baseline (elevated) blood pressure is recorded. Then each patient is given an experimental drug for a week and their blood pressure measured again to assess if the drug helped to reduce blood pressure, on average

• difference in two population means
• paired mean difference

8.b 1 point(s)

A long-term study evaluated the degree of addiction that teenagers form to nicotine once they begin experimenting with smoking. The study considered a random sample of 332 students (150 females, 182 males) who responded to the Hooked on Nicotine Checklist (HONC). The checklist included questions such as Have you ever tried to quit but couldnt? and Is it hard to keep from smoking in places you are not supposed to, like school? Each of ten such questions was answered using a scale of 1 = No problem, 2 = Small problem, 3 = Moderate problem, and 4 = Large problem. One primary response of interest is thetotal HONC score, defined to be the sum of the responses to the ten questions (so the minimum is 10 points to a maximum of 40 points). One research question is to assess if the average score differs for males versus females.

• difference in two population means
• paired mean difference

8.c 1 point(s)

Do Big 10 basketball players score more points at home games or at away games? A sports enthusiast took a random sample of 50 players across the Big 10. For each player, one home game and one away game were randomly selected and the number of points scored for each of these two games were recorded.

• difference in population means
• paired mean difference

Conscientiousness is a tendency to show self-discipline, act dutifully, and aim for achievement. The trait shows a preference for planned rather than spontaneous behavior. The Hogan Personality Inventory (HPI) is designed to be a measure of a persons level of conscientiousness.

Suppose the model for HPI scores for the population of all older adults aged 24-33 is a normal distribution with a mean of 155 points and a standard deviation of 20 points. Suppose the model for HPI scores for the population of all young adults aged 18-23 is a normal distribution with a mean of 145 points and a standard deviation of 15 points.

A random sample of 70 older adults aged 24-33 will be selected and administered the HPI to obtain a sample mean HPI score for these older adults. An independent random sample of 50 young adults aged 18-23 will also selected and administered the HPI to obtain a sample mean HPI score for these young adults. These two sample means will be compared by computing their difference, namely,.What is theapproximatedistribution of the possible values for such a difference in sample means?Be complete, provide all aspects of the distribution.