# Econometrics- Statistics

1. A study based on a sample size of 36 reported a mean of 87 with a margin of error of 10 for 95% confidence. (2 points)

a. Give the 95% confidence interval. (1 point) b. If you wanted 99% confidence for the same study, would your margin of error be greater than,

equal to, or less than 10? (1 point)

2. The Student Monitor surveys 1200 undergraduates from 100 colleges semiannually to understand trends among college students. Recently, the Student Monitor reported that the average amount of

time spent per week on the Internet was 19.0 hours. Assume that the standard deviation is 5.5 hours.

Give a 95% confidence interval for the mean time spent per week on the Internet. (1 point)

3. You want to rent an unfurnished one-bedroom apartment in Dallas next year. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $980. Assume the monthly

rents in Dallas follow a Normal distribution with a standard deviation of $290. (3 points)

a. Find a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartment available for rent in this community. (1 point)

b. Find a 98% confidence interval for the mean monthly rent for unfurnished one-bedroom apartment available for rent in this community. (1 point)

c. What can you say about the relationship between range of the confidence interval and the level of confidence? (1 point)

5. State the appropriate null hypothesis H0 and alternative hypothesis Ha in each of the following cases. (3 points)

a. A 2008 study reported that 88% of students owned a cell phone. You plan to take an SRS of students to see if the percentage has increased. (1 point)

b. The examinations in a large freshman chemistry class are scaled after grading so that the mean score is 75. The professor thinks that students who attend early morning recitation sections will

have a higher mean score than the class as a whole. Her students this semester can be considered

a sample from the population of all students she might teach, so she compares their mean score

with 75. (1 point)

c. The student newspaper at your college recently changed the format of their opinion page. You take a random sample of students and select those who regularly read the newspaper. They are

asked to indicate their opinions on the changes using a five-point scale: -2 if the new format is

much worse than the old, -1 if the new format is somewhat worse than the old, 0 if the new

format is the same as the old, +1 fi the new format is somewhat better than the old, and +2 if the

new format is much better than the old. (1 point)

6. Translate each of the following research questions into appropriate H0 and Ha. (2 points)

a. Census Bureau data show that the mean household income in the area served by a shopping mall is $42,800 per year. A market research firm questions shoppers at the mall to find out whether the

mean household income of mall shoppers is higher than that of the general population. (1 point)

b. Last year, your online registration technicians took an average of 0.4 hours to respond to trouble calls from students trying to register. Do this year’s data show a different average response time?

(1 point)

ECO 480 Econometrics I

Problem Set 3 (60 points)

Due: Wednesday, October 28, 2015 (beginning of the class)

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7. A test of the null hypothesis H0: 𝜇 = 𝜇0 gives test statistic z = 1.63. (3 points)

a. What is the P-value if the alternative is Ha: 𝜇 > 𝜇0? (1 point) b. What is the P-value if the alternative is Ha: 𝜇 < 𝜇0? (1 point) c. What is the P-value if the alternative is Ha: 𝜇 ≠ 𝜇0? (1 point)

8. Since previous studies have reported that elite athletes are often deficient in their nutritional intake (e.g., total calories, carbohydrates, protein), a group of researchers decided to evaluate Canadian high

performance athletes. A total of n = 324 athletes from eight Canadian sports centers participated in

the study. One reported finding was that the average caloric intake among the n = 201 women was

2403.7 kcal/day. The recommended amount is 2811.5 kcal/day. Is there evidence that female

Canadian athletes are deficient in the caloric intake? Assuming a standard deviation of 880 kcal/day,

carry out the appropriate test. Give the P-value and then interpret the result in one sentence using

plain language. (2 points)

9. Every user of statistics should understand the distinction between statistical significance and practical importance. A sufficiently large sample will declare very small effects statistically significant.

Consider the study of elite female Canadian athletes in question #8. Female athletes were consuming

an average of 2403.7 kcal/day with a standard deviation of 880 kcal/day. Suppose a nutritionist is

brought in to implement a new health program for these athletes. This program should increase mean

caloric intake but not change the standard deviation. Given the standard deviation and how caloric

deficient these athletes are, a change in the mean of 50 kcal/day to 2453.7 is of little importance.

However, with a large enough sample, this change can be significant. To see this, calculate the P-

value for the test of

H0: μ = 2403.7 Ha: μ > 2403.7

in each of the following situations (3points):

a. A sample of 100 athletes; their average caloric intake is �̅� = 2453.7. (1 point) b. A sample of 500 athletes; their average caloric intake is �̅� = 2453.7. (1 point) c. A sample of 2500 athletes; their average caloric intake is �̅� = 2453.7. (1 point)

10. Consider the study of elite female Canadian athletes in question #8. Now, a total of n =114 male athletes from eight Canadian sports centers were surveyed. The average caloric intake was 3077.0

kcal/day with a standard deviation of 987.0. The recommended amount is 3421.7. Is there evidence

that Canadian high performance male athletes are deficient in their caloric intake? (Hint: population

standard deviation is unknown.) (3 points)

a. State the appropriate H0 and Ha to test this. (1 point) b. Carry out the test, give the P-value, and state your conclusion. (1 point) c. Construct a 95% confidence interval for the average deficiency in caloric intake. (1 point)

11. Coronary heart disease (CHD) begins in young adulthood and is the fifth leading cause of death among adults aged 20 to 24 years. Studies of serum cholesterol levels among college students,

however, are very limited. A 1999 study looked at a large sample of students from a large

southeastern university and reported that the mean serum cholesterol level among women is 168

ECO 480 Econometrics I

Problem Set 3 (60 points)

Due: Wednesday, October 28, 2015 (beginning of the class)

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mg/dl with standard deviation of 27 mg/dl. The initial research was conducted to give a test of a

hypothesis about the cholesterol level for female sedentary undergraduates based on a sample of size

n = 71. The hypotheses are

H0: μ = 168 Ha: μ ≠ 168

While the result was not statistically significant, it did provide some evidence that the mean was larger

than 168. Thus, the researcher plans to recruit another sample of sedentary females but this time using

a one-sided alternative. He plans to obtain n = 70 subjects and wonders if this sample size give him

adequate power to detect an increase of 5 mg/dl to μ = 173. (4 points)

a. Given 𝛼 = 0.05, for what values of z will he reject the null hypothesis? (1 point) b. Using 𝜎 = 27 and 𝜇 = 168, for what values of �̅� will he reject H0? (1 point) c. Using 𝜎 = 27 and 𝜇 = 173, what is the probability �̅� will fall in the region defined in part (b)? (1

point)

d. Does it appear he has adequate power for a sample size of n = 70? Or does he need to find ways to increase it? In one sentence, explain your answer. (1 point)

12. An agronomist examines the cellulose content of a variety of alfalfa hay. Suppose that the cellulose content in the population has standard deviation 𝜎 = 8 milligrams per gram (mg/g). A sample of 15 cutting has mean cellulose content �̅� = 145 mg/g. (3 points)

a. Give a 90% confidence interval for the mean cellulose content in the population. (1 point) b. A previous study claimed that the mean cellulose content was 𝜇 = 140mg/g, but the agronomist

believes that the mean is higher than that figure. State H0 and Ha and carry out a significance test

to see if the new data support this belief. (1 point)

c. The statistical procedures used in parts (a) and (b) are valid when several assumptions are met. What are these assumptions? (1 point)

13. Traditional brand research argues that successful logos are ones that are highly relevant to the product they represent. However, a market research firm recently reported that nearly 20% of all table wine

brands introduced in the last three years feature an animal on the label. Since animals have little to do

with the product, why are marketers using this tactic? Some researchers have proposed that

consumers who are “primed” – in other words, they’ve thought about the image earlier in an unrelated

context – process visual information easier. To demonstrate this, the researchers randomly assigned

participants to either a primed or non-primed group. Each participant was asked to indicate their

attitude toward a product on a seven-point scale (from 1 = dislike very much to 7 = like very much).

A bottle of MagicCoat pet shampoo, with a picture of a collie on the label, was the product. Prior to

giving this score, however, participants were asked to do a word find where four of the words were

common across groups (pet, grooming, bottle, label) and four were either related to the image (dog,

collie, puppy, woof) or image conflicting (cat, feline, kitten, meow). Use BRANDPREFERENCE.csv

to answer the following questions. The dataset contains the responses listed from smallest to largest.

(9 points)

a. Examine the scores of each group graphically. (Hint: Draw a stemplot.) Is it appropriate to use the two-sample t procedures? Why or why not? Explain in one sentence. (3 points)

ECO 480 Econometrics I

Problem Set 3 (60 points)

Due: Wednesday, October 28, 2015 (beginning of the class)

5

b. Test whether these two groups show the same preference for this product. Use a two-sided alternative hypothesis and a significance level of 5%. (2 points)

c. Construct a 95% confidence interval for the difference in average preference. (2 points) d. What is your main finding? Summarize very briefly. (Maximum: 2 sentences) (2 points)

14. A recent study was performed to determine the prevalence of the female athlete triad (low energy availability, menstrual dysfunction, and low bone mineral density) in high school students. A total of

80 high school athletes and 80 sedentary students were assessed. The following table summarizes

several measured characteristics:

Athletes Sedentary

Characteristic �̅� s �̅� s

Body fat (%) 25.61 5.54 32.51 8.05

Boss mass index 21.6 2.46 26.41 2.73

Calcium deficit (mg) 297.13 516.63 580.54 372.77

Glasses of milk/day 2.21 1.46 1.82 1.24

For each of the characteristics, test the hypothesis that the means are the same in the two groups. Use a

significance level of 0.05 for each test.