# Mathematics

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STAT 100 Lesson 12 Assignment

Answer the following questions and submit for grading. Each question or part of a question is worth 1 point.

1. Researchers asked a sample of 50 1st grade teachers and a sample of 50 12th grade teachers how much of their own money they spent on school supplies in the previous school year. They wanted to see if teachers at one grade level spend more than teachers at the other grade level.

a. What type of study is found—observational or randomized experiment? Explain.

b. What is the experimental or observational unit?

c. What is the response variable?

d. What is the explanatory variable?

e. Do we have to worry about confounding variables in this instance? Why? If so identify a possible confounding variable?

f. Are either of the terms retrospective study or prospective study relevant? Explain.

2. A research team compared two methods of measuring tread wear on tires. Eleven tires were each measured for tread wear by two different methods: one method was based on weight while the other method was based on groove wear. For convenience, each tire was measured first by weight method and then second by the groove wear.

a. The two samples are which of the following: two independent or two dependent (matched pairs)? Explain.

b. What type of study is found—observational or randomized experiment? Explain.

c. What is the experimental or observational unit?

d. What is the response variable?

e. What is the explanatory variable?

3. A study wants to determine if taking fish oils can reduce depressive symptoms. A group of 50 volunteers who suffered from mild depression were randomly divided into two groups. Each person was given a three-month’s supply of capsules. One group was given capsules that contained fish oils while the other group was given capsules that look and tasted like fish oils, but actually only contained sugar. Neither the participants nor the investigator knew what type of capsule they were taking. At the end of the month, a psychologist evaluated them to determine if their depressive symptoms had changed. Therefore, we are comparing the “change in depressive symptoms” for individuals in two groups. Explain whether each of the following terms applies to this study.

a. observational study

b. randomized experiment

c. placebo

d. placebo effect

e. single-blind

f. double-blind

g. matched pairs (dependent samples)

h. block design

i. independent samples

j. explanatory variable (What is it?)

k. response variable (What is it?)

4. Does the use of cell phones lead to a higher incidence of brain cancer? People with brain cancer were matched with people who did not have brain cancer on age, gender, and living environment. Each participant in the study was asked to answer questions about previous life experiences and exposures. Determine whether or not each of the following terms applies to this observational study.

a. prospective

b. retrospective

c. case-control study

5. A study involving ten people wants to compare the effectiveness of two different brands of antihistamines with regard to enhancing sleep. Each person is randomly assigned to take Antihistamine A on one night and Antihistamine B on the other night. With each person, the hours of sleep were recorded for each night. Explain whether each of the following terms applies to this study.

a. observational study

b. randomized experiment

c. carry-over effect (confounding)

d. matched pairs (dependent samples)

e. explanatory variable (What is it?)

f. response variable (What is it?)

6. Suppose the study found in the previous problem instead found that each person took Antihistamine A on the first night and Antihistamine B on the second night. What terms that did not apply to the previous problem now apply to this problem? Explain.

7. Are you annoyed with spam e-mail? Suppose a random sample of 200 Penn State students was asked this question of which 80% said that they are annoyed. From the provided information we can find the following:

sample percent = 80% (sample proportion = .80)

standard deviation (S.D.) = .03

a. Set up the calculation of a 95% confidence interval to estimate the population proportion of Penn State students who are annoyed by spam e-mail? (Hint: refer to Example 12.5)

b. Knowing that the margin of error = .06 or 6%, write out a one-sentence interpretation of the margin of error.

c. The 95% confidence interval to estimate the population proportion of Penn State students who are annoyed by spam e-mail is (.74 to .86). Write out a one-sentence interpretation of this confidence interval.

d. What type of data is used in this example—categorical or measurement?

e. What would happen to the size of the margin of error or confidence interval if the level of confidence were instead 99.7%? Explain.

8. Explain what will happen to the width of a confidence interval (increase, decrease, or remain the same) as a result of each of the following changes:

a. Population size is doubled from 5 million to 10 million

b. Confidence level lowered from 98% to 90%

c. Sample size is doubled from 500 to 1000

9. A sample of 200 students in a Stat class were asked “How long did you sleep last night?” The results are found below.

sample mean = 6.4 hours

S.D. = 1.6 hours

standard error (S.E.) = .11 hours

sample size (n) = 200

a. Set up the calculation of a 95% confidence interval to estimate the population mean number of hours slept last night.

b. Knowing that the margin of error = .22, write out a one-sentence interpretation of the margin of error.

c. The 95% confidence interval to estimate the population mean number of hours slept last night is (6.18 to 6.62) hours. Write out a one-sentence interpretation of this confidence interval.

d. What type of data is used in this example?

e. What would happen to the size of the margin of error or confidence interval if the level of confidence were instead 68%? Explain

10. A 95% confidence interval for the proportion of women that have ever dozed off while driving is 0.07 to 0.14. For men, a 95% confidence interval for the proportion that have ever dozed off while driving is 0.19 to 0.25. Assume both intervals were computed using large random samples.

a. What conclusion can be made about the two population proportions that have dozed off while driving? Why?

b. Rewrite each confidence interval in terms of percents rather than proportions. Does the conclusion remain the same? Explain.

c. The two samples are which of the following: two independent or two dependent (matched pairs)?

d. What type of data is found in this problem—categorical or measurement?

e. Would the conclusion remain the same if the two confidence intervals had instead been calculated at 90% confidence? Explain.

11. Attention Deficient Hyperactivity Disorder (ADHD) is a diagnosis applied to children who exhibit the following behaviors: (1) inattention (2) impulsiveness, (3) hyperactivity. ADHD is now known to be a lifelong problem where adolescents and adults continue to exhibit symptoms. Researchers at the University of Wisconsin (Heiligenstein E. et al., 1999) explored both psychological and academic functioning in ADHD college students. They reviewed charts of students who voluntarily sought a comprehensive assessment at the University’s Counseling and Consultation Services. Relevant charts were classified into two groups:

· Control Group: 28 students who requested a career interest inventory but did not receive or request any counseling sessions beyond those needed for the career inventory

Students in both groups completed the Inventory of Common Problems (ICP). The ICP is an established self-report measure (inventory) of college student problems that includes 31 questions in seven subset areas. The two subset areas that we will examine are (1) Academic Problems and (2) Depression. In each subset area, there were four questions each where a rating of 1 to 5 was possible. Because of this, within each subset area, the minimum score was 4 points and the maximum score was 20 points.

a. What type of observational study did the researchers at the University of Wisconsin use? Explain.

b. The two samples are which of the following: two independent or two dependent (matched pairs)?

c. Table 1 provides the results for the subset area: Academic Problems. What conclusion can be made when comparing the ADHD group to the control group? Why? Can you conclude that being ADHD causes a student to have a higher score in the subset area of academic problems? Explain.

Table A1. Results for Subset Area: Academic Problems (Heiligenstein E. et al, 1999)

 ADHD Controls Sample Size (n) 26 28 Mean Score 14.5 points 10.4 points St Dev (S.D.) 3.7 points 3.9 points S.E. .73 points .74 points 95% C.I. for Population Mean Subset Score 14.5 ± 2(.73) = 14.5 ± 1.5 = approx (13 to 16 ) points 10.4 ± 2(.74) = 10.4 ± 1.5 = approx (9 to 12) points

d. Table 2 provides the results for the subset area: Depression. What conclusion can be made when comparing the ADHD group to the control group? Why?

Table A2. Results for Subset Area: Depression (Heiligenstein E. et al, 1999)

Sample Size (n) 26 28
Mean Score 8.3 points 7.0 points
St Dev (S.D.) 2.5 points 3.2 points
###### S.E
.5 points .6 points
95% C.I. for Population Mean Subset Score 8.3 ± 2(.5) =

8.3 ± 1.0 = approx

(7 to 9) points

7.0 ± 2(.6) =

7.0 ± 1.2 = approx

(6 to 8) points

12. Are low carbohydrate diets effective? A random sample of six individuals who wanted to try a low carbohydrate diet was obtained. Each individual was placed on a low carbohydrate diet for eight weeks. The weight in pounds was determined for each individual both before and after the diet, as shown in Table A3.

a. The two samples are which of the following: two independent or matched pairs? Explain.

b. What type of study is found—observational or randomized experiment? Explain.

c. What is the experimental or observational unit?

d. What is the response variable?

e. What is the explanatory variable?

f. What sample(s) are used to calculate the appropriate confidence interval?

g. The following information was obtained from the table found below.

sample mean difference = 13.2 pounds

S.D. = 13 pounds

standard error (S.E) = 5.2 pounds

sample size (n) = 6 people

Set up the calculation for a 95% confidence interval to estimate the population mean difference.

h. Suppose the 95% confidence for the population mean difference in weight is (2.8 to 23.6) pounds. What conclusion can be made in this instance about the effectiveness of the diet? Explain in statistical terms.

i. What is the advantage of using the differences rather than the original data in the calculation of the confidence intervals?

Table A3. Weight Before and After Diet

 Person Weight Before Diet (pounds) Weight After Diet (pounds) Difference in pounds = (Before-After) 1 125 117 8 2 165 151 14 3 205 169 36 4 115 117 -2 5 138 132 6 6 152 135 17

13. Two methods of memorizing difficult material are being tested to determine if one method produces better retention. Nine pairs of students are included in the study. Each student in the pair has been matched according to IQ and academic background and then randomly assigned to use one of the two methods: Method A or Method B. A memorization test is given to all the students where the final score can range from 0 to 100 points.

Table A4. Memorization Methods

 Sample 95% C.I. for Population Mean Score on Memorization Test Method A (50 to 74) points Method B (45 to 73) points Difference = (Method A – Method B) (1 to 5) points

a. The two samples are which of the following: two independent or matched pairs? Explain.

b. What type of study is found—observational or randomized experiment? Explain.

c. What is the experimental or observational unit?

d. What is the response variable?

e. What is the explanatory variable?

f. What confidence interval(s) from Table A4 should be used to compare the memorization score for the two methods? Explain.

g. Using your answer in the previous part as your basis, state an appropriate conclusion in terms of the problem.