Question

There is a general belief that students drink more and more coffee as they spend more time in college. To test this belief, a student organization randomly selects 37 freshmen and 32 seniors at the U of I. The amount of daily coffee consumption of each student was recorded. For the freshmen, the sample average is 8.4 with a standard deviation of 3.4. For the seniors, the sample average is 10.2 with a standard deviation of 2.7.

Assume that the samples are independent and that the population standard deviations of the amount daily coffee consumption of freshmen and seniors at the U of I are the same.

Conduct a hypothesis test at significance level = 0.02.

Which of the following is the correct ? (Note that we define difference = senior - freshmen)

(a) Both sample average = 0

(b) The difference in the population averages = 0

(c) The difference in the sample averages = 0

(d) Both population averages = 0

Choose the correct alternative hypothesis .

(a) The difference in the population averages  0

(b) The difference in the population averages > 0

(c) The difference in the sample averages  0

(d) The difference in the population averages < 0

Compute the test statistic for the test that you chose above.

Test statistic:

Make sure your answer contains exactly 2 digits after the decimal point.

Use your rounded answer for question 3 and the table to compute the p-value.

p-value:

Make sure your answer contains at least 4 digits after the decimal point.

Which of the following is the correct conclusion?

(a) We reject the null hypothesis and concludes there is no evidence that on average, U of I seniors drink's daily coffee consumption is more than that of freshmen.

(b) We reject the null hypothesis and concludes that on average, U of I seniors drink's daily coffee consumption is more than that of freshmen.

(c) We fail to reject the null hypothesis and concludes there is no evidence that on average, U of I seniors drink's daily coffee consumption is more than that of freshmen.

(d) We fail to reject the null hypothesis and concludes that on average, U of I seniors drink's daily coffee consumption is more than that of freshmen.

Answer 1

HYPOTHESIS TEST-

Suppose, random variables X and Y denote amount of daily coffee consumption of senior and freshman respectively.

Here, two different groups are used to collect data. Further we do not know population standard deviation (or variance). So, we have to perform two sample t-test.

We have to test for null hypothesis

against the alternative hypothesis

Our test statistic is given by

Here,

Senior sample size

Freshman sample size

Sample mean for senior

Sample mean for freshman

Pooled sample standard deviation is given by

Degrees of freedom

[Using R-code '1-pt(2.40848,67)']

Level of significance

We reject our null hypothesis if

Here, we observe that

So, we reject our null hypothesis.

Hence, based on the given data we can conclude that there is significant evidence that seniors consume more coffee than freshmen.

ANSWERS-

Null hypothesis is (b) The difference in the population averages = 0.

Alternative hypothesis is (b) The difference in the population averages > 0.

Test statistic is t = 2.41.

p-value = 0.0094

We conclude as follows.

(b) We reject the null hypothesis and conclude that on average, U of I seniors drink's daily coffee consumption is more than that of freshmen.