1) An urn contains 10 balls, 2 red, 5 blue, and 3 green balls. Take out 3 balls at a random, without replacement. You win $2 for each green ball you select and lose $3 for each red ball you select. Let the random variable X denote the amount you win, determine the probability mass function of X.

2) Each of the 60 students in a class belongs to exactly one of the three groups A,B, or C. The membership numbers for the 3 groups are as follows:

A: 15 students

B: 20 C: 25 First, Next,

a. b. c.

students
students
choose one of the 60 students at random and let X be the size of that student’s group;.

choose one of the 3 groups at random and let Y be its size. Write down the pmf for X and Y.
Compute ?(?) ??? ?(?).
Compute ???(?) ??? ???(?).

3. Roll a fair die repeatedly. Let X denote the number of 4’s in the first 6 rolls and let Y denote the number of rolls needed to obtain a 2.

a. WritedowntheprobabilitymassfunctionofX. b. WritedowntheprobabilitymassfunctionofY. c. Find an expression for ?(? ≥ 3).
d. Find an expression for ?(? > 6).

More than 100 million people around the world are not getting enough sleep; the average adult needs between 7.5 and 8 hours of sleep per night. College students are particularly at risk of not getting enough shut-eye. A recent survey of several thousand college students indicated that the total hours of sleep time per night, denoted by the random variable X, can be approximated by a normal model with E(X) = 6.84 hours and SD(X) = 1.2 hours.

Question 1. Find the probability that the hours of sleep per night for a random sample of 4 college students has a mean x between 6.6 and 6.91. (use 4 decimal places in your answer)

Question 2. Find the probability that the hours of sleep per night for a random sample of 16 college students has a mean x between 6.6 and 6.91. (use 4 decimal places in your answer)

Question 3. Find the probability that the hours of sleep per night for a random sample of 25 college students has a mean x between 6.6 and 6.91. (use 4 decimal places in your answer)

A small community college claims that their average class size is equal to 35 students. This claim is being tested with a level of significance equal to 0.02 using the following sample of class sizes: 42, 28, 36, 47, 35, 41, 33, 30, 39, and 48. Assume class sizes are normally distributed. (NOTE: We need to assume that class sizes are normally distributed in order to use the t distribution because the sample size n=10 < 25, and the Central Limit Theorem does not apply.)

Which of the following conclusions can be drawn?

• A.

  Since the test statistic equals 1.36, fail to reject the null hypothesis and conclude that there's insufficient evidence to conclude that class size does not equal 35 students.

• B.

  Since the test statistic equals 2.26, fail to reject the null hypothesis and conclude that class size does equal 35 students.

• C.

  Since the test statistic equals 2.05, reject the null hypothesis and conclude that class size does not equal 35 students.

• D.

  Since the test statistic equals 1.58, reject the null hypothesis and conclude that class size does not equal 35 students.

During a blood-donor program conducted during finals week for college students, a blood-pressure reading is taken first, revealing that out of 300 donors, 44 have hypertension. All answers to three places after the decimal.

The probability, at 60% confidence, that a given college donor will have hypertension during finals week is? , with a margin of error of?

Assuming our sample of donors is among the most typical half of such samples, the true proportion of college students with hypertension during finals week is between?. and?

We are 99% confident that the true proportion of college students with hypertension during finals week is ? , with a margin of error of ?. .

Assuming our sample of donors is among the most typical 99.9% of such samples, the true proportion of college students with hypertension during finals week is between ? and ?

Covering the worst-case scenario, how many donors must we examine in order to be 95% confident that we have the margin of error as small as 0.01? Using a prior estimate of 15% of college-age students having hypertension, how many donors must we examine in order to be 99% confident that we have the margin of error as small as 0.01? .

1. In a study examining whether taking a statistics course influences students’ reasoning skill, a researcher first gave a reasoning task to 5 students who have never taken a statistics course, and then the students took a brief statistics lecture which lasted 3 hours. After the lecture, the same five students took the reasoning task again. Their performance scores are summarized in the table below.

Perform an appropriate statistical test and provide your conclusion on whether or not their reasoning performance are different before and after taking the STATS lecture.

2. In the above (#9), the same 5 students took the test twice, before and after the lecture. Reanalyze the above data as the data obtained from two sets of independent samples. Now please perform an appropriate statistical test to compare the means, pretending that two different (independent) groups of five people took the same test, one before and one after the lecture.

1. A salsa producer has just received a shipment of tomatoes from their main supplier. If the salsa
company finds convincing evidence that more than 7% of the tomatoes are damaged and
unusable for the salsa making process, the truck will have to be sent away and a new shipment
will have to be sent out by the supplier. An inspection reveals that 49 tomatoes are unusable
when the supervisor selects a random sample of 500 tomatoes from the truck. Carry out a
significance test at the α = 0.05 significance level. What should the salsa producer conclude
about the shipment?

2. As part of the Southeastern Region of Teach America, researchers conducted two surveys in
2019. The first survey asked a random sample of 1287 Florida college students about their use of
online tutoring services. A second survey posed similar questions to a random sample of 1354
Georgia college students. In these 2 studies, 57% of Florida college students and 68% of Georgia
college students said they used online tutoring services. Construct and interpret a 90%
confidence interval for the difference between the proportion of Florida and Georgia college students who use online tutoring services

4. One hundred students were interviewed. Forty–two are Monthly Active Users (MAU) of Facebook (F), and sixty–five are MAU of Snapchat(S). Thirty–four are MAU of both Facebook and Snapchat. One of the100 students is randomly selected, all 100 students having the same probability of selection (1/100).

(a) What is the probability that the student is an MAU of Facebook?

(b) What is the probability that the student is an MAU of Facebook given that the student is an MAU of Snapchat?

(c) Find Pr(S|F).

d) Express the probability in Part (b) in symbols, that is in a similar fashion to Part (c).

5. The students in Question 4 were also asked if they were MAU of Twitter. Twenty five were MAU of Twitter, including 15 who were MAU of Facebook and Twitter and 16 who were MAU of Twitter and Snapchat. Twenty four students were not MAU of any of Facebook, Snapchat or Twitter.

(a) What is the probability that a randomly selected student is an MAU of Facebook, Snapchat, and Twitter?

(b) What is the probability that a randomly selected student is an MAU of Twitter, conditional on that student being an MAU of both Facebook and Snapchat?

FCAT scores and poverty. In the state of Florida, elementary school performance is based on the average score obtained by students on a standardized exam, called the Florida Comprehensive Assessment Test (FCAT). An analysis of the link between FCAT scores and sociodemographic factors was published in the Journal of Educational and Behavioral Statistics (Spring 2004). Data on average math and reading FCAT scores of third graders, as well as the percentage of students below the poverty level, for a sample of 22 Florida elementary schools are summarized by the number given below. (x= percentage of students below poverty level, and y=math score ) n = 22 ??xi = 1292.7 ??yi = 3781.1 ??x2i =88668 ??yi2 =651612 ??xiyi =218292 (a) Propose a straight-line model relating math-score to percentage of students below poverty level. (b) Find the least-squares regression line fitting the model to the data. (c) Interpret the estimates for intercept and slope in the context of the problem. (d) Test whether the math score is negatively related to the percentage of students below the poverty level. (e) Construct a 99% confidence interval for the slope of the model, and interpret your result in the context of the problem.

1. As part of their class requirements, the students in Dr. Taylor’s Research Design and Analysis class are sent over to Trumbull Mall to observe interactions between mothers and their toddler-aged children. They are told not to interact with the moms at all, but just record certain behaviors, like the number of times they speak harshly to their children and the number of times the children whine or cry.

One of the mothers notices that the students are watching people and complains to mall security. The manager of the mall asks the students where they are from, then writes a letter of complaint to Dr. Taylor. Here is an excerpt:

”I am requesting that you do not engage in any more observational research at Trumbull Mall. I don’t think it is right to allow students to observe people’s behavior without getting their permission first. It is a violation of privacy and is wrong even if they don’t realize they are being watched. People come to the mall to shop, not to be watched.”

Reflection Questions:

1. What are some good reasons for the manager’s concerns? Explain.
2. What are some good reasons why the students should be able to do this type of research? Explain.
3. If you were in Dr. Taylor’s position, how would you handle the situation?