Question

Assignment 3: Hypothesis Testing for the Population Mean

The purpose of this activity is to give you guided practice in going through the process of a t-test for the population mean, and teach you how to carry out this test using statistical software.

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?

Frequency 35+ 30- 25+ 20- 15- 10 5- 20 10 15 number of drinks per week un LO O N II

Instructions

Excel Instructions: Hypothesis Testing for the Population Mean

To open Excel with the data in the worksheet, click here and it will automatically download the file to your computer. Find the downloaded file (usually in the Downloads folder) and double-click it to open it in Excel.

To carry out the test using Excel, we need to find the value of the test statistic (t), which we calculate using

Before we can do this, we first have to calculate the values for each of the variables in the equation:

• First, we need to calculate x-bar, the sample mean. To do this, we use the AVERAGE function in Excel—in this case, =AVERAGE(A2:A76).
• Next, we calculate s, the sample standard deviation, using the STDEV function in Excel over the same range of data.
• We know μ_o and n from reading the problem statement, so we can now calculate a value for t.
• Finally, we find our P-value using the Excel function TDIST(t, n-1, 2).

Question 3:

State the test statistic, interpret its value and show how it was found.

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

Question 1.

The claim “ 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 ” can be written as $\mu\neq 4.73$. Its complement is $\mu=4.73$ because $\mu=4.73$ contains the statement of equality, it becomes the null hypothesis. We can write the null and alternative hypotheses as shown

$H_{0}:\mu=4.73$

$H_{a}:\mu\neq4.73$

Question 2.

From the histogram, we observe that the distribution is right shewed and having outliers also. But the sample size is enough large (n=75). Using central limit theorem, we apply the test and ensure that sample mean is normal. So, t-test can be used.

Question 3. From the histogram, we find the data series

Form the excel, we calculate sample mean and sample standard deviation is

$n=75, \bar{x}=3.9333, s= 3.7752$

Test Statistic is

$t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}=\frac{3.93-4.73}{\frac{3.78}{\sqrt{75}}}=-1.827$

Question 4 For p-value, the degree of freedom is n-1 =74 and level of significance is 5%.

Here, the hypothesis is two sided tailed, so

$p-value=0.0717$

The p-value is greater than 0.05. Do not reject the null hypothesis . This concluded that mean number of alcoholic drinks that students at his college have in a typical week differs from 4.73, the mean of U.S. college students.

Question 5. Here, hypothesis is converted to one sided test. SO, the p-value is reduce to its half, i.e., p-value = 0.0358

At 5% level of significance, we reject the null hypothesis because p-value is less than 0.05. This concluded that mean number of alcoholic drinks that students at his college have in a typical week lower than the mean of U.S. college students

Question 6. Yes, we would still be mathematically justified in using the T-test for obtaining conclusions, as we did previously because the sample size is not change, only the background of the student is change. it is not affect the test statistic.