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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.