On February 24^th, 2020, MU University Board of Visitors announced that Dr.Washington was selected as the university’s eighth president. In a Washington Post article, Dr. Washington mentioned that he was a first-generation college student. It is known that 39% of MU students are the first in their families to attend college. A random sample of ten STAT 250 students were asked the question, “Are you a first-generation college student?”

1. Check if this situation fits the binomial setting. Write four complete sentences addressing each requirement in one sentence each.

a. Assuming this situation is a binomial experiment, build the probability distribution in table form.  use the binomial calculator and calculate the probability of each of the values of the random variable from X = 0 to X = 10. present this table horizontally or vertically and leave the probabilities unrounded.

b. Calculate the probability that exactly four of the students in this sample are first-generation college students using the binomial calculator. Then, write one sentence to interpret the probability in context of the question.

c. Calculate the probability that at least two students in this sample are first-generation college students using the probability distribution table you created in (a). Show your work “by hand.” Then, verify your answer using the binomial calculator.

d. Calculate the probability that between four and seven (inclusive) of the students in this sample are first-generation college students the binomial calculator graph and include this image with values.

e. What is the average number of students you expect to respond “yes” to being a first-generation student? To answer this question, calculate the mean and standard deviation of this probability distribution. Show your work using the binomial mean and binomial standard deviation formulas and provide your answers. Round to two decimal place when necessary.

f. Imagine you repeated this sample of ten students 10,000 times. Produce a properly titled and labeled relative frequency bar plot.

g. Compare the height of the bar above four with your answer to part (b) and identify which type of probability each value is.

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)?

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Thirty randomly selected students took the calculus final. If the sample mean was 75 and the standard deviation was 7.2​, construct a​ 99% confidence interval for the mean score of all students.

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