Question

DATA

3
8
2
15
2
2
0
0
4
5
2
7
0
1
5
3
0
2
5
4
1
6
9
5
3
1
2
10
6
1
1
2
1
19
6
6
6
7
0
4
1
1
1
0
1
9
2
2
2
1
16
10
10
5
2
3
1
4
4
4
3
6
2
8
5
2
7
1
6
4
0
3
1
1
1

Background:

A group of 75 college students from a certain liberal arts college were randomly sampled and asked about the number of alcoholic drinks they have in a typical week. The file containing the data is linked below. The purpose of this study was to compare the drinking habits of the students at the college to the drinking habits of college students in general. In particular, the dean of students, who initiated this study, would like to check whether the mean number of alcoholic drinks that students at his college have in a typical week differs from the mean of U.S. college students in general, which is estimated to be 4.73.

Question 1:

Let μ be the mean number of alcoholic beverages that students in the college drink in a typical week. State the hypotheses that are being tested in this problem.

Question 2:

Here is a histogram of the data. Can we safely use the t-test with this data?

Instructions

Click on the link corresponding to your statistical package to see instructions for completing the activity, and then answer the questions below.

R | StatCrunch | Minitab | Excel 2007 | TI Calculator

Question 3:

State the test statistic and interpret its value.

Question 4:

Based on the P-value, draw your conclusions in context.

Question 5:

What would your conclusions be if the dean of students suspected that the mean number of alcoholic drinks that students in the college consume in a typical week is lower than the mean of U.S. college students in general? In other words, if this were a test of the hypotheses:

H₀: μ = 4.73 drinks per week

H_a: μ < 4.73 drinks per week

Question 6:

Now suppose that instead of the 75 students having been randomly selected from the entire student body, the 75 students had been randomly selected only from the engineering classes at the college (for the sake of convenience).

Address the following two issues regarding the effect of such a change in the study design:

a. Would we still be mathematically justified in using the T-test for obtaining conclusions, as we did previously?

b. Would the resulting conclusions still address the question of interest (which, remember, was to investigate the drinking habits of the students at the college as whole)?

Answer 1

Solution-1:

Ho:

Ha:

Soluiton-b:

df1 =read.table(header = TRUE, text ="
drinks
3
8
2
15
2
2
0
0
4
5
2
7
0
1
5
3
0
2
5
4
1
6
9
5
3
1
2
10
6
1
1
2
1
19
6
6
6
7
0
4
1
1
1
0
1
9
2
2
2
1
16
10
10
5
2
3
1
4
4
4
3
6
2
8
5
2
7
1
6
4
0
3
1
1
1
"
)
df1

histogram(df1$drinks,main='Histogram')

From histogram we could see that shape is not symmetrical

shape is positively skewed

But as for central limt theorem,n=75

sample follows normal distribution

State the test statistic and interpret its value.

Solution-3:

Rcode to get

test statistic and p value using t.tets function

Rcode:

t.test(df1$drinks,mu=4.73)

Output:

data: df1$drinks
t = -1.8275, df = 74, p-value = 0.07165
alternative hypothesis: true mean is not equal to 4.73
95 percent confidence interval:
3.064735 4.801932
sample estimates:
mean of x
3.933333

test statitstic,t=-1.8275

Solution-4:

p-value = 0.07165

p>0.05

Fail to reject Ho

Accept Ho

Conclusion;
There is no suffcient statistical evidence at 5% level of signficance to conclude that the mean number of alcoholic drinks that students at his college have in a typical week differs from the mean of U.S. college students in general, which is estimated to be 4.73.

Solution-5:

R code;

t.test(df1$drinks,mu=4.73,alternative="less")

Output;

data: df1$drinks
t = -1.8275, df = 74, p-value = 0.03583
alternative hypothesis: true mean is less than 4.73
95 percent confidence interval:
-Inf 4.659456
sample estimates:
mean of x
3.933333

t=-1.8275

, p-value = 0.03583

p<0.05

Reject Ho

Accept Ha

Comclusion:

There is sufcient statistical evidence at 5% level of significance to conclude that mean number of alcoholic drinks that students in the college consume in a typical week is lower than the mean of U.S. college students in general.