1. John made an experiment by tossing three fair coins. (Fair coin has the same probability for a tail and head ½).
(a) (3 points) List the sample space for this experiment. (All possible outcomes)
(b)(2 point) What is a probability of three tails?
(c) (2 points) What is a probability of exactly two tails?
(d) (3 points) What is a probability of at least one tail?
2. Among UMUC students 70% own a car, 50% own a bike and 40% own both.
(a) (4 points) Draw a Venn diagram.
(b) (4 points) Find a probability that a randomly chosen student own a least one of the above vehicles.
(c) (4 points) What is a probability that a randomly chosen student does not own a bike?
3. A division has 12 employees, 8 males and 4 females.
(a) (3 points) In how many ways can a committee of 6 members can be selected from out of these 12?
(b) (4 points) Find the probability that a random committee contains all males?
(c) (3 points) Suppose that the committee should have one President
which has to be a female and two ordinary members who must be males.
How many different committees are possible?
4. X is normally distributed with mean 12 and standard deviation 4.
(a) (3 points)Find the probability that X will be more than 10.
(b) (4 points) Find the probability that X will be less than 13.5
(c) (5 points) Find the probability that X will be between 10 and 14.
(d) (5 points) Find a such that P(X<a)=.85
(e)(5 points) Find b such that P(X> b)=.25
5. (4 points) Which of the following is not a binomial distribution and why? Justify your answer.
1. Tossing a fair quarter 10 times and looking at number of heads that shows up.
2. Rolling a fair die 10 times and looking at the number of times we get six dots showing up.
3. Rolling a fair die 10 times and keeping track of the numbers that are rolled.
4. Rolling 10 fair dice and looking at the number of dice that have 4 dots facing up.
6. Among companies doing highway or bridge construction, 80% test employees for substance abuse (based on data from the Construction Financial Management Association) A study involves the random selection of 12 such companies.
(a) (5 points) Find the probability that exactly 7 of the 12 companies test for substance abuse.
(b) (4 points) Find the probability that at least half of the companies test for substance abuse.
(c) (9 points) For such group of 12 companies, find the mean and standard deviation for the number (among 12) that test for substance abuse.
(d) (4points) Using the results from part (c) and the range rule of thumb, indentify the range of usual value.
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 get- ting better each trial? Test the linear effect of trial for the data.
a. Compute L for each subject using the contrast weights -1, 0, and 1. That is, compute (-1)(a) + (0)(b) + (1)(c) for each subject. b. Compute a one-sample t-test on this column (with the L values for each subject) you created.
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. Ho: x ¯ = 4.5, Ha : x ¯ > 4.5
b. Ho: μ ≥ 4.5, Ha: μ < 4.5
c. Ho: μ = 4.75, Ha: μ > 4.75
d. Ho: μ = 4.5, Ha: μ > 4.5
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?
80. Your statistics instructor claims that 60 percent of the students who take her Elementary Statistics class go through life feeling more enriched. For some reason that she can’t quite figure out, most people don’t believe her. You decide to check this out on your own. You randomly survey 64 of her past Elementary Statistics students and find that 34 feel more enriched as a result of her class. Now, what do you think?
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
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.
Mean score before class
Mean score after class
The correct decision is:
a. Reject H0.
b. Do not reject the H0