Probability and Statistics Exam: Questions and Answers

Probability questions

Please type or write-in the answers in the appropriate box. There are a total of 50 points possible. The exam is open note and open book. You cannot collaborate with others. Please be sure to put your name on the exam so that you can get credit for your work.

A bag of M&M has six colors of M&Ms: blue, red, green, yellow, brown, and orange. If the bag has an equal number of each of the six colors, what are the probabilities for each of the following?

	Answer	Points Possible
Randomly selecting a yellow candy?		1
Randomly selecting either a blue, red, or orange candy?		1
Randomly selecting something other than a yellow or brown candy?		1

	Answer	Points Possible
p (z < 0.40)		1
p (z > -0.25)		1
p (0.30 < z < 1.60)		1

	Answer	Points Possible
What is the percentile rank for X = 75?		1
What is raw score for the 11^th percentile?		1
What scores comprise Q₁ and Q₃, and what is the interquartile range?		1

	Answer	Points Possible
What is the probability that the sample mean will be greater than 52?		1
What is the probability that the sample mean will be less than 54?		1
What is the probability that the sample mean will be within 4 points of the population mean?		1

Based on the following research questions, indicate whether you should conduct a z test, a single sample t test, an independent samples t test, or a related samples t test?

	Type of Test	Points Possible
A researcher randomly sampled male and female youth to participate in a delinquency prevention program. She measured the youth’s attitudes about antisocial behavior before and after participating in the program.		1
A researcher was interesting is studying differences in policing in poor and wealthy communities. The researcher gathered information on a sample of poor and wealthy communities and measured the number of disorderly conduct arrests in each community.		1
A researcher wanted to compare the average recidivism rate between a sample of juvenile programs and the population of juvenile programs. The researcher has the population mean but not the population standard deviation.		1

A professor at University of Alabama conducted a study on the relationship between type of legal council (public defender vs. private) and the length of sentence received by the client. The researcher found that there is a statistically significant relationship between type of council and length of sentence. Using this information answer the following questions.

	Answer	Points Possible
What type of error might the research be making?		1
Which type of error is considered worse in the scientific community (and the book)?		1
Provide one strategy you could use to reduce the likelihood that you might reject a null hypothesis that is true?		1

Answer

Points

Possible

A researcher examines whether an over-the-counter cold medication influences mental alertness of prison wardens. A random sample of n = 16 participants is obtained, and each person is given a standard dose of the medication one hour before being tested on a simulation task. For the general population, scores on the simulation task are normally distributed with µ = 60 and σ = 8. The individuals in the sample had an average score of M = 56.

	Answer	Points Possible
What is the population?		1
What is the sample?		1
Does this study meet the assumption of scale of measurement? Why or why not?		2
Does this study meet the assumption of sampling distribution is normal? Why or why not?		2
For this study what is the null hypothesis?		1
Can the researcher conclude that scores on the driving simulation task are significantly different after taking the medication? Use a two-tailed test with α = 0.05		1
Using the two-tailed test with α = 0.05 compute Cohen’s d and interpret the size of the effect.		1
Can the researcher conclude that scores on the driving simulation task are significantly lower after taking the medication? Use a one-tailed test with α = 0.01.		1

A professor at the University of Wisconsin is interested in studying the relationship between officer’s gender and citizen complaints. The professor gathered information on 55 officers randomly sampled from the Chicago Police Department. Using the data collected below apply the five steps of the hypothesis testing process to answer the research question: is there a relationship between gender and citizen complaints. Use a two-tailed test with an alpha level of 0.05.

Male Officers	Female Officer
n = 35	n = 20
M = 10.57	M = 8.34
SS = 901.34	SS = 794.34

	Answer	Points Possible
Step 1. State hypotheses and alpha level.		3
Step 2. Locate critical region.		1
Step 3. Collect data and calculate test statistic.		1
Step 4. Make a decision about the null and research hypothesis.		2
Compute and interpret Cohen’s d effect size.		1
Compute the 95% confidence interval for mean difference		1
Based on your decision regarding step 4 above what type of error might you be making?		1
Write up the results for the t-test following American Psychological Association (APA) guidelines.		1

A researcher at the Oregon State was interested in studying the problem of burnout among Black officers. In the first part of his study, he compared levels of burnout between Black and White officers (n = 37). However, he was concerned that data might violate the assumption of homogeneity.

Black Officers	White Officers
n = 17	n = 20
Range = 0-6	Range = 0-6
M = 1.76	M = 2.58
s² = 7.08	s² = 10.45
s = 2.66	s = 3.23

		Points Possible
Explain the assumption of homogeneity.		1
Calculate the F-max statistic		1
Do the results of the F-max test indicate that the data violate the assumption of homogeneity?		1

A researcher would like to evaluate the effectiveness of a treatment program designed to reduce aggressiveness in domestic violence offenders. Prior to participating in the program, each of n = 8 participants has their aggressiveness rated on a scale from 1 to 10 by a clinical psychologist. After participating in the program, a second aggressiveness rating is recorded. The data are below. Using this information answer the questions below.

Before	After
6	2
8	3
9	4
8	1
10	2
5	3
9	8
7	7

	Answer	Points Possible
Compute the mean for the sample of difference scores (M_D).		1
Compute the estimate for the standard error (		1
Compute the t statistic.		1
Do the results indicate that the program has a statically significant effect?		1
Compute r² and interpret the size of the effect.		1
Write up the results of the t test following the American Psychological Association (APA) guidelines.		1

Answer

To calculate the probability that the sample mean will be greater than 52, we need to know the distribution of the population and the sample size.

Assuming that the population is normally distributed and the sample size is sufficiently large (i.e., greater than 30), we can use the central limit theorem to approximate the distribution of the sample mean as normal with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Let's say we know the population mean is 50 and the population standard deviation is 10. If we take a sample of size 100, then the distribution of the sample mean can be approximated as normal with mean 50 and standard deviation 1 (i.e., 10 divided by the square root of 100).

To calculate the probability that the sample mean will be greater than 52, we need to standardize this value using the formula:

z = (x - μ) / (σ / sqrt(n))

Get instant help from 5000+ experts for