Exercise 7 Confidence intervals and tests for proportions
Alcohol abuse may be a problem for some university students. In a survey in the US from the 1990s, a number of students were asked about their drinking behavior and alcohol related problems. In particular the investigators were interested in the proportion of the students who were "frequent binge drinkers", defined as having had five or more drinks in a row three or more times the last two weeks. Among the 9916 women who were interviewed, 1684 were classified as frequent binge drinkers. A total of 7180 males took part in the study, and among them 1630 were frequent binge drinkers.
a) Estimate the proportion of frequent binge drinkers among male and female students. Also give 95% confidence intervals for these proportions.
We want to investigate if there is a difference in the drinking habits among male and female students.
b) Estimate the difference in the proportions of frequent binge drinkers among males and females. Also give a 95% confidence interval for the difference in proportions.
c) Use the z-test described on slide 8 of Lecture 6 to test the null hypothesis that there is no difference between male and female students when it comes to frequent binge drinking. What do you conclude from the test?
An alternative to the z-test used in question c is to use a chi-squared test for a 2x2 table; cf. slides 10-11. Such a test is based on a comparison of observed (O) and expected (E) numbers.
d) The 2x2 table below give the observed numbers of male and female students who are and aren't frequent binge drinkers, but two numbers are not given in the table. Fill in the numbers that are missing.
|
Observed numbers (O's) |
Frequent binge drinkers |
Not frequent binge drinkers |
Total |
|
Males |
1630 |
|
7180 |
|
Females |
|
8232 |
9916 |
|
Total |
3314 |
13782 |
17096 |
e) The 2x2 table below give the expected numbers of male and female students who are and aren't frequent binge drinkers (assuming that there is no difference between the genders), but two numbers are not given in the table. Explain how the expected numbers are computed and fill in the two missing numbers.
|
Expected numbers (E's) |
Frequent binge drinkers |
Not frequent binge drinkers |
Total |
|
Males |
|
5788.2 |
7180 |
|
Females |
|
7993.8 |
9916 |
|
Total |
3314 |
13782 |
17096 |
f) The chi-squared test statistic is a sum of terms of the form (O-E)2/E over the four cells of the 2x2 table. The 2x2 table below give these terms, but one of them is not given in the table. Fill in the missing number and use the table to compute the chi-squared statistic. What can you conclude from the test?
|
(O-E)2/E |
Frequent binge drinkers |
Not frequent binge drinkers |
|
Males |
40.77 |
9.80 |
|
Females |
|
7.10 |
g) What is the relation between the z-test statistic in question c and the chi-squared test statistic in question f ?