1. Ten different senators are randomly selected without replacement, and the number of terms that they have served are recorded. Does this constitute a binomial distribution? Select an answer, and then state why.
No
Yes
Why: This is not a binomial distribution because each “trial” (the selection of a senator) has more than two possible outcomes.
2. Which of the following pairs are NOT independent events?
Flipping a coin and getting a head, then flipping a coin and getting a tail
Throwing a die and getting a 6, then throwing a die and getting a 5
Selecting a red marble from a bag, returning the marble to the bag, then selecting a blue marble
Drawing a spade from a set of poker cards, setting the card aside, then selecting a diamond from the set of poker cards
All of the above are independent events
3. Exam scores from a previous STATS 200 course are normally distributed with a mean of 74 and standard deviation of 2.65. Approximately 95% of its area is within:
One standard deviation of the mean
Two standard deviations of the mean
Three standard deviations of the mean
Depends on the number of outliers
Must determine the z-scores first to determine the area
4. You had no chance to study for the final exam and had to guess for each question. The instructor gave you three choices for the final exam:
I: 10 questions, each question has 5 choices, must answer at least 4 correct to pass
P(x ≥ 4) = 1 – P(x = 0) – P(x = 1) – P(x = 2) – P(x = 3)
II: 5 questions, each question has 4 choices, must answer at least 3 correct to pass
P(x ≥ 3) = 1 – P(x = 0) – P(x = 1) – P(x = 2)
III: 4 questions, each question has 6 choices, must answer at least 2 correct to pass
P(x ≥ 2) = 1 – P(x = 0) – P(x = 1)
Which final exam format offers the highest probability to pass?
Final exam I
Final exam II
Final exam III
All three final formats have equal probabilities
Need more information to compute probabilities
5. Consider a normal distribution with a mean of 12 and variance of 4. Approximately 82% of the area lies between which values?
a. 6 and 13
b. 10 and 16
c. 9 and 15
d. 10 and 18
e. Not enough information provided to solve
6. For a standard normal distribution, what's the probability of getting a number
less than zero?
a. 75%
b. 63%
c. 50%
d. 43%
e. 34%
7. Which description of normal distributions is correct (select all that apply)?
Normal distributions have a mean of zero and standard deviation of one.
Normal distributions can differ in their means, but their standard deviations must be the same.
Standard normal distributions cannot differ in both their means and their standard deviations.
Normal distributions cannot differ in their means, but can differ in their standard deviations.
None of the above are correct
8. Consider an extremely right skewed distribution with a mean of 15 and standard deviation of 2. 99.7% of its area is within:
One standard deviation of the mean
Two standard deviations of the mean
Three standard deviations of the mean
2.5 standard deviations of the mean
Can't determine from the information given.
9. A delivery truck must make stops in eight different cities, designated by the first letter in the name of the city: A, B, C, D, E, F, G, and H. If the order in which the truck visits the eight locations is chosen randomly, what is the probability that the truck will visit them in reverse alphabetical order?
a. b. c. d.
10. Acme Airlines flies airplanes that seat 100 passengers. From experience, they have determined, on average, 84% of the passengers holding reservations for a particular flight actually show up for the flight. If they book 116 passengers for a flight, what is the probability (rounded to four decimals) that 100 or fewer passengers holding reservations will actually show up for the flight?
a. 0.8400 b. 0.8590 c. 0.8621 d. 0.7774 e. 0.7241
11. A jar contains 12 marbles, 5 of which are green and 7 of which are blue. If 2 marbles are chosen at random (without replacement) and then 2 additional marbles are chosen at random (without replacement), what is the probability of selecting 3 green marbles and 1 blue marble?
12. If events A and B are mutually exclusive events, each with non-zero probability then which of the following is true:
a. P(A n B) = P(A) + P(B)
b. P(A u B) = P(A) + P(B)
c. P(A) – 1 = P(B)
d. P(A) = P(B)
e. P(A n B) = P(A) * P(B)
13. An elevator has a stated maximum capacity of 12 people or 2004 pounds. If 12 people have weights with a mean greater than (2004/12) = 167 pounds, the capacity will be exceeded. Assume that weights of men are normally distributed with a mean of 182.9 pounds and a standard deviation of 40.8 pounds. Show your work and round your answers to FOUR decimal places.
a. Compute the probability that a randomly selected man will have a weight greater than 167 pounds.
b. Compute the probability that 12 randomly selected men will have a mean weight that is greater than 167 pounds.
c. Does the elevator appear to have the correct weight limit? Why or why not?
14. A company has initiated a training program for new hires. After surveying 25 new employees, they determined the average training time was 7.5 hours with a sample standard deviation of 2.25 hours. Assume that the underlying population is normally distributed. Show your work and round your CI to FOUR decimal places.
a. Define the random variable X for this problem in words.
b. Define the random variable for this problem in words.
c. Construct a 95% confidence interval for the population mean length of time of new hire training.
d. A new employee scheduled for the training program, stated he would only need 6 hours to complete the training. Is his claim reasonable? State why or why not.
15. A researcher randomly surveyed 300 high school seniors and determined 225 stated they drive a car to high school. We are interested in the population proportion of seniors who drive a car to high school.
a. Define the random variable X for this problem in words.
b. Define the random variable P' for this problem in words.
c. Construct a 90% confidence interval (CI) for the population proportion of high school seniors who claim to drive a car to high school. Round your CI to FOUR decimal places.
d. Is it reasonable to conclude at least 80% of seniors drive a car to high school?
18. You choose an alpha level of .01 and then analyze your data.
a. What is the probability that you will make a Type I error given that the null hypothesis is true?
b. What is the probability that you will make a Type I error given that the null hypothesis is false?
20. True/false: It is easier to reject the null hypothesis if the researcher uses a
smaller alpha (α) level.
7. Below are data showing the results of six subjects on a memory test. The three scores per subject are their scores on three trials (a, b, and c) of a memory task. Are the subjects getting better each trial? Test the linear effect of trial for the data.
a
|
b
|
c
|
4
|
6
|
7
|
3
|
7
|
8
|
2
|
8
|
5
|
1
|
4
|
7
|
4
|
6
|
9
|
2
|
4
|
2
|
a. compute L for each subject using the contrast weights -1, 0, and 1. That is, compute (-l)(a) + (O)(b) + (l)(c) for each subject.
b. compute a one-sample t-test on this column (with the L values for each subject) you created.
c. In words, CLEARLY state what your random variable or P' represents.
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p-value? In one or two complete sentences, explain what the p-value means for this problem.
g. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
h. Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
i. Construct a 95% confidence interval for the true mean or proportion. Include a sketch of the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval.
13. You are conducting a study to see if students do better when they study all at once or in intervals. One group of 12 participants took a test after studying for one hour continuously. The other group of 12 participants took a test after studying for three twenty minute sessions. The first group had a mean score of 75 and a variance of 120. The second group had a mean score of 86 and a variance of 100.
a. What is the calculated t value? Are the mean test scores of these two groups significantly different at the .05 level?
b. What would the t value be if there were only 6 participants in each group? Would the scores be significant at the .05 level?
65. Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test.
The null and alternative hypotheses are:
a.
b.
c.
d.
71. Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test, the Type I error is:
a. to conclude that the current mean hours per week is higher than 4.5, when in fact, it is higher
b. to conclude that the current mean hours per week is higher than 4.5, when in fact, it is the same
c. to conclude that the mean hours per week currently is 4.5, when in fact, it is higher
d. to conclude that the mean hours per week currently is no higher than 4.5, when in fact, it is not higher
77. An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% level?
Hypothesis Testing with One Sample
a. Ho:___
b. Ha:___
c. In words, CLEARLY state what your random variable or P' represents.
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p-value? In one or two complete sentences, explain what the p-value means for this problem.
g. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
h. Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
i. Alpha: _ _ _
ii. Decision: _ _ _
iii. Reason for decision: _ _ _
iv. Conclusion: _ _ _
i. Construct a 95% confidence interval for the true mean or proportion. Include a sketch of the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval.
80. At Rachel's 11th birthday party, eight girls were timed to see how long (in seconds) they could hold their breath in a relaxed position. After a two-minute rest, they timed themselves while jumping. The girls thought that the mean difference between their jumping and relaxed times would be zero. Test their hypothesis.
Hypothesis Testing with Two Samples
a. H0:___
b. Ha:___
c. In words, CLEARLY state what your random variable , ,
or represents.
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p-value? In one or two complete sentences, explain what the p-value means for this problem.
g. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
h. Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
i. Alpha: _ _ _
ii. Decision: _ _ _
iii. Reason for decision: _ _ _
iv. Conclusion: _ _ _
i. In complete sentences, explain how you determined which distribution to use.
91. A powder diet is tested on 49 people, and a liquid diet is tested on 36 different people. Of interest is whether the liquid diet yields a higher mean weight loss than the powder diet. The powder diet group had a mean weight loss of 42 pounds with a standard deviation of 12 pounds. The liquid diet group had a mean weight loss of 45 pounds with a standard deviation of 14 pounds.
Hypothesis Testing with Two Samples
a. H0:___
b. Ha:___
c. In words, CLEARLY state what your random variable , ,
or represents.
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p-value? In one or two complete sentences, explain what the p-value means for this problem.
g. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
h. Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
i. Alpha: _ _ _
ii. Decision: _ _ _
iii. Reason for decision: _ _ _
iv. Conclusion: _ _ _
i. In complete sentences, explain how you determined which distribution to use.
120. A golf instructor is interested in determining if her new technique for improving players' golf scores is effective. She takes four new students. She records their 18-hole scores before learning the technique and then after having taken her class.
She conducts a hypothesis test. The data are as follows.
The correct decision is:
a. Reject Ho.
b. Do not reject the Ho.
a. H0:___
b. Ha:___
c. In words, CLEARLY state what your random variable , ,
or represents.
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p-value? In one or two complete sentences, explain what the p-value means for this problem.
g. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
h. Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
i. Alpha: _ _ _
ii. Decision: _ _ _
iii. Reason for decision: _ _ _
iv. Conclusion: _ _ _
i. In complete sentences, explain how you determined which distribution to use.
In the North American court system, a defendant is assumed innocent until proven guilty. In an ideal world, we would expect that the truly innocent will always go free, whereas the truly guilty ones will always be convicted. Now, let us tackle the following questions?
1. In the context of the Type I error and Type II error, can you relate a court trial scenario in terms of these two errors?
What would be your ideal situation if you are the defendant?
What would be your ideal situation if you are the prosecuting attorney?
Lastly, what do you think of the scenario of an ideal world where we expect that no innocent will be found guilty and all guilty will be convicted in the context of Type I error and Type II error?
