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Frequency Distribution, Confidence Interval and Hypothesis Tests

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1. The frequency distribution below was constructed from data collected on the quarts of soft drinks consumed per week by 20 students.

Quarts of Soft Drink Frequency
0 - 3 4
4 - 7 5
8 - 11 6
12 - 15 3
16 - 19 2

a. Construct a relative frequency distribution.
b. Construct a cumulative frequency distribution.
c. Construct a cumulative relative frequency distribution.

2. A researcher has obtained the number of hours worked per week during the summer for a sample of fifteen students.

40 25 35 30 20 40 30 20 40 10 30 20 10 5 20

Using this data set, compute the
a. median
b. mean
c. mode
d. 40th percentile
e. range
f. sample variance
g. standard deviation

3. Assume you have applied to two different universities (let's refer to them as Universities A and B) for your graduate work. In the past, 25% of students (with similar credentials as yours) who applied to University A were accepted, while University B accepted 35% of the applicants. Assume events are independent of each other.

a. What is the probability that you will be accepted in both universities?
b. What is the probability that you will be accepted to at least one graduate program?
c. What is the probability that one and only one of the universities will accept you?
d. What is the probability that neither university will accept you?

4. The probability distribution for the rate of return on an investment is

The Rate of Return
(In Percent) Probability
9.5 .1
9.8 .2
10.0 .3
10.2 .3
10.6 .1

a. What is the probability that the rate of return will be at least 10%?
b. What is the expected rate of return?
c. What is the variance of the rate of return?

5. Scores on a recent national statistics exam were normally distributed with a mean of 80 and a standard deviation of 6.

a. What is the probability that a randomly selected exam will have a score of at least 71?
b. What percentage of exams will have scores between 89 and 92?
c. If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an award?
d. If there were 334 exams with scores of at least 89, how many students took the exam?

6. The average weekly earnings of bus drivers in a city are $950 with a standard deviation of $45. Assume that we select a random sample of 81 bus drivers.

a. Compute the standard error of the mean.
b. What is the probability that the sample mean will be greater than $960?
c. If the population of bus drivers consisted of 400 drivers, what would be the standard error of the mean?

7. A university planner is interested in determining the percentage of spring semester students who will attend summer school. She takes a pilot sample of 160 spring semester students discovering that 56 will return to summer school.

a. Construct a 95% confidence interval estimate for the percentage of spring semester students who will return to summer school.
b. Using the results of the pilot study with a 0.95 probability, how large of a sample would have to be taken to provide a margin of error of 3% or less?

8. A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days. The standard deviation of the population () is known to be 5.6 days.

a. State the null and alternative hypotheses.
b. Using the critical value approach, test to determine if their advertisement is legitimate. Let a = .05.
c. Using the p-value approach, test the hypotheses at the 5% level of significance.

9. In order to estimate the difference between the average Miles per Gallon of two different models of automobiles, samples are taken and the following information is collected.

Model A Model B
Sample Size 60 55
Sample Mean 28 25
Sample Variance 16 9

a. At 95% confidence develop an interval estimate for the difference between the average Miles per Gallon for the two models.
b. Is there conclusive evidence to indicate that one model gets a higher MPG than the other? Why or why not? Explain.

10. In order to determine whether or not a driver's education course improves the scores on a driving exam, a sample of 6 students were given the exam before and after taking the course. The results are shown below.
Let d = Score After - Score Before.

Score Score
Student Before the Course After the Course
1 83 87
2 89 88
3 93 91
4 77 77
5 86 93
6 79 83

a. Compute the test statistic.
b. At 95% confidence using the p-value approach, test to see if taking the course actually increased scores on the driving exam.

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Solution Summary

The solution gives detailed steps on solving a set of questions on the frequency distribution, confidence interval and hypothesis testing using either t distribution or normal distribution.

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Frequcny distribution chart and confidence interval

Scenario:
You are an associate assigned to the claims department of a major insurance company. A policy holder has had an accident with his classic 1968 Oldsmobile Cutlass Supreme.

At issue is the consideration of his brakes. Recently, he was driving down the road and was apparently unable to stop in time when a woman driving a 2004 Porsche Boxster "S" pulled out in front of him; subsequently, he "T-boned" the Porsche. Injured in the accident were the, driver of the Porsche, who was 7 months pregnant with twins and her elderly mother-in-law who has cerebral palsy; both of which were from out of town. Now with their only means of transportation "totaled" they are stranded.

The Porsche driver's insurance company, which is USA, contends that your policy holder is at fault because his car was not up to current standards. There seems to be a difference in the braking distance between vintage brake shoes and current ones.

Your policy holder is a self proclaimed, "shade tree" mechanic and a classic car enthusiast. As a matter of fact, he once owned a rather successful auto mechanic business and is now the President of the State of Florida Oldsmobile Club which has a substantial lobby in Tallahassee.

Your boss decided that because you are enrolled in a statistics class you should be pressed into service to assist and, as such, you have questioned the policy holder extensively. From your investigation you discover that he does his own work and recently replaced his brakes with a vintage brand of asbestos brake shoes. The contention is that modern brake shoes stop a vehicle which is traveling at 35 mph (which your policy holder was pr oven to be doing) at 20.5 feet, give or take one foot either side.

Your company's research department gathered the following in support:
Out of 42 sets available, worldwide, of vintage asbestos brake shoes, 20 were selected for testing. Below are the results:
23.2 18.1 19.2 20.3 23.0
26.0 24.6 16.9 17.3 23.4
28.6 17.2 23.2 18.7 19.6
20.8 24.2 25.0 19.8 17.6

Questions;
A. What is the percentage of these pads that fall within the current and
more modem parameters?
B. Arrange this data in class intervals and construct a Frequency
Distribution chart.
C. Construct a confidence interval to predict the boundaries of this
parameter.
D. Your policy holder actually contends that the vintage brake shoes exceed the modem standards. Is there evidence that would suggest he is correct?
E. Assuming that the stopping distance between the vintage brake shoe and the modem equivalent is the same, what is the probability that either one tested will stop a vehicle of this size or larger, within 5 feet of the true mean?
F. What is the probability that the difference between the stopping. distance with the vintage brake pads and the stopping distance of the newer style brake pads being as large as reported or larger if there is no difference in the true stopping distance averages between the two styles?

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