Frequency Distribution, Confidence
Interval and Hypothesis Tests
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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