Exam 6: Accessing Associations Between Variables

An elementary school principal wants to know the mean number of children in families whose Children attend this school. He checks all the families using the school's registration records, and we Create a 95% confidence interval based on a t-distribution. This procedure was not appropriate. Why?

A random sample of 120 classrooms at a large university found that 70% of them had been cleaned Properly. What is the standard error of the sample proportion?

Which statement correctly compares t-distributions to the normal distribution? I. t distributions are also mound shaped and symmetric. II. t distributions have less spread than the normal distribution. III. As degrees of freedom increase, the variance of t distributions becomes smaller.

Trainers need to estimate the level of fat in athletes to ensure good health. Initial tests were based On a small sample but now the trainers double the sample size for a followup test. The main Purpose of the larger sample is to…

Based on data from two very large independent samples, two students tested a hypothesis about Equality of population means using $\alpha = 0.02$ 2) One student used a one-tail test and rejected the null Hypothesis, but the other used a two-tail test and failed to reject the null. Which of these might Have been their calculated value of t?

A soft drink company is conducting research to select a new design for the can. A random sample of participants has been selected. Instead of a typical taste test with two different sodas, they actually give each participant the same soda twice. One drink is served in a predominantly red can, the other in a predominantly blue can. The order is chosen randomly. Participants are asked to rate each drink on a scale of 1 to 10. Thus, the company wishes to test if the color of the can influences the rating. The ratings were recorded for each participant. The data are shown in the table below. Does this sample indicate that there is a difference in the ratings? Test an appropriate hypothesis and state your conclusion. Rater Red Blue 1 4 6 2 7 5 3 3 6 4 8 9 5 5 2 6 9 9 7 7 10 8 5 4 9 6 8 10 9 7 11 8 8 12 3 7 13 6 5 14 8 8 15 9 10 16 7 6

Gas mileage Hoping to improve the gas mileage of their cars, a car company has made an adjustment in the manufacturing process. Random samples of automobiles coming off the assembly line have been measured each week that the plant has been in operation. The data from before and after the manufacturing adjustments were made are in the table. It is believed that measurements of gas mileage are normally distributed. Write a complete conclusion about the manufacturing adjustments based on the statistical software printout shown below.

A professor was curious about her students' grade point averages (GPAs). She took a random Sample of 15 students and found a mean GPA of 3.01 with a standard deviation of 0.534. Which of The following formulas gives a 99% confidence interval for the mean GPA of the professor's Students?

Scrubbers A factory recently installed new pollution control equipment ("scrubbers") on its smokestacks in hopes of reducing air pollution levels at a nearby national park. Randomly timed measurements of sulfate levels (in micrograms per cubic meter) were taken before (Set C1) and after (Set C2) the installation. We believe that measurements of sulfate levels are normally distributed. Write a complete conclusion about the effectiveness of these scrubbers based on the statistical software printout shown.

Based on data from two very large independent samples, two students tested a hypothesis about Equality of population means using $\alpha = 0.02$ 2) One student used a one-tail test and rejected the null Hypothesis, but the other used a two-tail test and failed to reject the null. Which of these might Have been their calculated value of t?

A marketing company reviewing the length of television commercials monitored a random sample Of commercials over several days. They found that a 95% confidence interval for the mean length (in seconds) of commercials aired daily was (23, 27). Which is true?

The two samples whose statistics are given in the table thought to come from populations with Equal variances. What is the pooled estimate of the population standard deviation? n Mean SD 50 22 3 55 25 4

A professor runs a regression to see how students' exam scores (Y) are related to their homework Grades (X). The R2 of the regression is 21%. What does R2 tell us?

Test identification Suppose you were asked to analyze each of the situations described below. (NOTE: Do not do these problems!) For each, indicate which procedure you would use (pick the appropriate number from the list), the test statistic ( $z$ , $t$ , or $\chi ^ { 2 }$ "chi-squared"), and, if $t$ or $\chi ^ { 2 }$ , the number of degrees of freedom. A choice may be used more than once. 1. proportion - 1 sample 2. difference of proportions - 2 samples 3. mean - 1 sample 4. difference of means - independent samples 5. mean of differences - matched pairs 6. goodness of fit 7. homogeneity 8. independence a. Which takes less time to travel to work-car or train? We select a random sample of 45 businessmen, observe them commuting using both methods, and compare their travel times. b. A college professor wonders if two versions (A and B) of his exam are equally difficult. He randomly mixes 20 Version A's and 20 Version B's together and passes them out to his 40 students. After grading the exams, he compares the scores for the two versions. c. Forty people complaining of allergies take an antihistamine. They report that their discomfort subsided in an average of 18 minutes; the standard deviation was 4 minutes. The manufacturer wants a 95% confidence interval for the "relief time". d. A health professional selected a random sample of 100 patients from each of four major hospital emergency rooms to see if the major reasons for emergency room visits are similar in all four major hospitals. The major reason categories are accident, illegal activity, illness, or other. e. A policeman believes that about 40% of older drivers speed on highways, but a confidential survey found that 49 of 88 randomly selected older drivers admitted speeding on highways at least once. Is this strong evidence that the policeman was wrong? f. According to United Nations Population Division, the age distribution of the Commonwealth of Australia is: 21% less than 15 years of age, 67% between 15 and 65 years of age, and 12% are over 65 years old. A random sample of 210 residents of Canberra revealed 40 were less than 15 years of age, 145 were between 15 and 65 years of age, and 25 were over 65 years old. Is Canberra unusual in any way? g. Among a random sample of college-age students, 6% of the 473 men said they had been adopted, compared to only 4% of the 552 women. Does this indicate a significant difference between adoption rates of males and females in college-age students?

One common method of evaluating the performance of a mutual fund is to compare its returns to those of a recognized benchmark such as an index of the returns on all securities of the type that the fund accumulates. The Janus Worldwide Fund considers its benchmark to be the MSCI World IndexSM. The table below depicts the annual returns (percent) for a recent ten-year period. Is this fund a good investment? That is, does this fund significantly outperform its benchmark? Source: https://ww3.janus.com/advisor/Documents/Advisor%20Lit%20System/Fact%20Sheets/4Q12%20Fact%20Sheet%20(Janus%20Worldw ide%20Fund-Class%20A)_exp%2004-15-13.pdf -Carry out the appropriate test and state your conclusion in context.

Blood pressure Researchers developing new drugs must be concerned about possible side effects. They must check a new medication for arthritis to be sure that it does not cause an unsafe increase in blood pressure. They measure the blood pressures of a group of 12 subjects, then administer the drug and recheck the blood pressures one hour later. The drug will be approved for use unless there is evidence that blood pressure has increased an average of more than 20 points. They will test a hypothesis using $\alpha = 0.05$ a. Write appropriate hypotheses (in words and in symbols). b. In this context, which do you consider to be more serious - a Type I or a Type II error? Explain briefly. c. After this experiment produced inconclusive results the researchers decided to test the drug again another group of patients. Describe two changes they could make in their experiment to increase the power of their test, and explain the disadvantages of each.

The vast majority of states and the District of Columbia have adopted the Common Core State Standards (CCSS) for math and English language arts. Do teachers support the CCSS? In March 2003, The American Federal of Teachers (AFT) asked AFT member teachers "Based on what you know about the Common Core State Standards and the expectations they set for children, do you approve or disapprove of your state's decision to adopt them? " The following results were reported in American Educator (Volume 32, No. 2, Summer 2013, pg. 3): 27% Strongly Approve; 48% Somewhat Approve; 14% Somewhat Disapprove; 8% Strongly Approve; 3% Not Sure. A district superintendent asked the same question to the teachers in her district to assess the level of teacher support for the CCSS within the district. She obtained the following results. Response Strongly Approve Somewhat Approve Somewhat Disapprove Strongly Disapprove Not Sure Frequency 55 106 28 32 9 a. Test an appropriate hypothesis to ascertain if the district CCSS approval distribution matches the national AFT approval distribution. b. Which response impacted your decision the most? Explain what this means in the context of the problem.

A total of 23 Gossett High School students were admitted to State University. Of those students, 7 were offered athletic scholarships. The school's guidance counselor looked at their composite ACT scores (shown in the tabl, wondering if State U. might admit people with lower scores if they also were athletes. Assuming that this group of students is representative of students throughout the state, what do you think? -Test an appropriate hypothesis and state your conclusion.

College admissions According to information from a college admissions office, 62% of the students there attended public high schools, 26% attended private high schools, 2% were home schooled, and the remaining students attended schools in other countries. Among this college's Honors Graduates last year there were 47 who came from public schools, 29 from private schools, 4 who had been home schooled, and 4 students from abroad. Is there any evidence that one type of high school might better equip students to attain high academic honors at this college? Test an appropriate hypothesis and state your conclusion.

A random sample of 120 classrooms at a large university found that 70% of them had been cleaned Properly. What is the standard error of the sample proportion?