U6500 Homework #8 | Complete Solution

Quantitative Analysis – International and Public Affairs
U6500 Sections 1, 2, 3, 4/Fall 2015

Homework #8: Inference for one mean, Difference in two means, one proportion,
and Difference in two proportions

Due in Mailbox 49 6th Floor IAB:
Wednesday November 11 4:00 p.m. (Sections 3 & 4)/
Thursday November 17 4:00 p.m. (Sections 1 & 2)
52 points total

1.    (8 points total)  Bags of a certain brand of tortilla chips claim to have a net weight of 14 oz.  Net weights actually vary slightly from bag to bag. Assume net weights are Normally distributed.  A representative of a consumer advocate group wishes to see if there is any evidence that the mean net weight is less than advertised and so intends to test the hypotheses H0:  = 14 versus Ha:  < 14.  To do this, he selects 16 bags of tortilla chips of this brand at random and determines the net weight of each.  He finds a sample mean of 13.88 oz. with a sample standard deviation of 0.24 oz.  
a.    What is the value of the test statistic?
b.    What is the critical value at the .05 significance level for this test?  Infer the p-value from the table.
c.    What is your conclusion (again, assume significance level of .05)?   Briefly explain your conclusion (citing your observed test statistic and p-value and what you are comparing these against).

2.    (8 points total)      A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city.  The researcher obtained a simple random sample of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be six hours with a standard deviation of three hours.  The researcher also obtained an independent simple random sample of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be four hours with a standard deviation of two hours.  Let 1 and 2 represent the mean amount of time spent in extracurricular activities per week by the populations of all high school students in the suburban and city school districts, respectively.  Assume two-sample t procedures are safe to use.
a.    What is a 95% confidence interval for 1 – 2? Interpret your confidence interval in words.
b.    Based on your answer in part (a), would you reject the null hypothesis that the mean number of hours spent on extracurricular activities per week is equal between the two groups versus the two-sided alternative at the 5% significance level?  Briefly explain.  

3.    (5 points total)  A simple random sample of 85 students is taken from a large university on the West Coast to estimate the proportion of students whose parents bought a car for them when they left for college. When interviewed, 51 students in the sample responded that their parents bought them a car.
a.    Find the proportion of students whose parents bought a car for them when they left for college and the standard error.
b.    Give a 95% confidence interval for the true proportion of students from this University whose parents bought a car for them when they left for college.

4.    (11 points total)  An inspector inspects large truckloads of potatoes to determine the proportion p in the shipment with major defects prior to using the potatoes to make potato chips.  If there is clear evidence that this proportion is less than 0.10, she will accept the shipment.  To reach a decision, she will test the hypotheses H0: p = 0.10 versus Ha: p < 0.10.  The results must be significant at the .05 level for her to accept the shipment.  To do so, she selects a simple random sample of 150 potatoes from the more than 3000 potatoes on the truck.  Only 8 of the potatoes sampled are found to have major defects.
a.    Carry out the significance test.  What can you conclude? Make sure to report the test statistic and the p-value (and clearly state what you are comparing them with).  
b.    Should the inspector accept the shipment? Briefly explain.

STATA Problems: Use “Sesame Street Data Homework 8” dataset available on Courseworks
Description of Sesame Street Data

The television series Sesame Street is concerned mainly with teaching preschool skills to children age 3-5, with special emphasis on reaching economically disadvantaged children.  The show is designed to hold young children’s' attention through action-oriented, short duration presentations teaching specific preschool cognitive skills and some social skills. Each show is one hour and involves much repetition of concepts within and across shows.  

Does Sesame Street help economically disadvantaged children 'catch-up' with economically advantaged children?  In the early 1970s, researchers at Educational Testing Service (the company that runs the SAT) ran a study to evaluate Sesame Street.   The researchers sampled children representative of economically advantaged and disadvantaged populations from five different sites in the United States.  To ensure the study contained a group of children that watched Sesame Street regularly, they randomly assigned children either to receive encouragement to watch Sesame Street or not to receive encouragement.  Those assigned to encouragement were given promotional materials, and received weekly visits and phone calls from ETS staff.  Those assigned not to receive encouragement did not get this attention.

The children were tested on a variety of cognitive variables, including knowledge of body parts, knowledge about letters, knowledge about numbers, etc., both before and after viewing the series. The names of variables in the Sesame Street dataset are shown in the code book at the bottom of this assignment. Variable names starting with "pre" indicate the pre-treatment variables, and "post" indicates a post-treatment variable.
1)    (9  points total)  These are data from a randomized experiment so we expect that pre-treatment variables will have the same distributions across treatment groups.  You want to test that the means of the pretest score on body parts (prebody) across the two treatment groups (viewenc) are not significantly different from each other.
a.    Run the appropriate STATA command.  Show output.  
b.    State the null and alternative hypotheses (assume a two-sided test).  Briefly explain why this alternative hypothesis makes the most sense here.  
c.    Test these hypotheses at the 5 percent significance level using the appropriate p-value (and make sure you clearly indicate what you are comparing it against).

2)    (11  points total)      You will now test if encouragement (the treatment) led to an increase in post-test score for body parts (postbody).  
a.    Make a case for choosing either pooled or un-pooled estimates of the standard error for this test. Then run the STATA command accordingly.  Show output.
b.    Test the hypothesis that there is no difference in mean post-test scores (assume two-sided test and unequal variances) at the 5 percent significance level. Make sure you clearly cite the observed t-value and p-value (and what you are comparing them with).  
c.    The STATA output in part (a) can also allow us to interpret the "treatment effect."  Provide a brief statement about how much the treatment of being encouraged to watch Sesame Street improved children's average scores on the body part test administered after treatment.
 
Code book with variable names

id :         subject identification number

sex           male=1, female=2

age           age in months

viewcat      frequency of viewing
                  1=rarely watched the show
                  2=once or twice a week
                  3=three to five times a week
                  4=watched the show on average more than 5 times a week

setting:        setting in which Sesame Street was viewed, 1=home 2=school

viewenc :      treatment condition    1=child encouraged to watch,  2=child not encouraged             to watch

prebody :      pretest on knowledge of body parts (scores range from 0-32)

prelet :      pretest on letters (scores range from 0-58)

preform :     pretest on forms (scores range from 0-20)

prenumb :     pretest on numbers (scores range from 0-54)

prerelat :     pretest on relational terms (scores range from 0-17)

preclasf :     pretest on classification skills

postbody :     posttest on knowledge of body parts (0-32)

postlet :      posttest on letters (0-58)

postform :      posttest on forms (0-20)

postnumb :     posttest on numbers (0-54)

postrelat :     posttest on relational terms (0-17)

postclasf:      posttest on classification skills

peabody:      mental age score obtained from administration of the Peabody Picture Vocabulary         test as a pretest measure of vocabulary maturity