Download CENTRAL MICHIGAN UNIVERSITY

Page 1 of 6
CENTRAL MICHIGAN UNIVERSITY
Department of Mathematics
STA 282
Spring 2004
SUPPLEMENTARY EXAMPLES (not to be graded)
1.
(a)
(b)
(c)
(d)
2.
(a)
(b)
(c)
(d)
(e)
3.
(a)
(b)
(c)
(d)
4.
(a)
(b)
The following are the grades which 30 students obtained in a Calculus test:
75 77 83 88 74 94 85 75 91 87 76 70 62 73 50
90 79 82 69 66 65 98 58 77 89 87 56 63 81 71
The minimum and the maximum scores are underlined.
Using an appropriate lower boundary for the first class and a class size of 5 (i.e. let k = 5),
construct a frequency table for the given data.
Calculate the relative frequency of each class interval.
Plot the relative frequency histogram.
What proportion of the students scored 80 and above?
Suppose the following are the number of empty seats on 9 different flights from Lansing to
Chicago:
8, 9, 12, 3, 2, 11, 18, 8, 1
Calculate the sample mean.
Find the sample median.
Calculate the sample variance and hence the sample standard deviation.
How many measurements lie between (both end points inclusive) one standard deviation of
the mean?
Compare your result in (d) to the Empirical rule.
Consider the following sample consisting of eleven measurements.
10, 13, 15, 13, 19, 15, 13, 17, 17, 13, 9
Calculate the sample mean.
Find the sample mode.
Calculate the sample standard deviation.
What proportion of the measurements lie between (both end points inclusive) one standard
deviation of the mean?
Suppose A and B are events such that P(A) = 0.54, P(B) = 0.60, and P(A ∩ B) = 0.25
Compute:
(i) P(A’ ∩ B)
(ii) P(A|B)
(iii) P(A ∪ B)
(iv) P(A’ ∩ B’)
(v) P(A’ ∪ B)
Are A and B mutually exclusive? Justify your answer.
Page 2 of 6
5.
(a)
(b)
(c)
(d)
For married couples living in a certain suburb, the probability that the husband will vote in
a school election is 0.21, the probability that the wife will vote in the election is 0.28, and
the probability that they will both vote is 0.15.
What is the probability that at least one of them will vote?
What is the probability that exactly one of them will vote?
If the wife votes, what is the probability that the husband will vote?
Given that the husband has not voted, what is the probability that the wife will not vote?
6.
Consider the following probability distribution
x
p(x)
1
2
3
4
5
.15 .20 .30 .19 .16
(e)
Find:
(i) P(X ≥ 2)
(ii) P(3 < X ≤ 5)
(iii) P(X < 3).
Compute E(X).
Find the variance of X and hence the standard deviation of X.
Find the fraction of X that lies between (both end points inclusive) one standard deviation
of the mean.
Compare your result in (d) with the Empirical rule.
7.
Consider the probability function
(a)
(b)
(c)
(d)
x
p(x)
1
2
3
4
.04 .20 .30 .46
(a)
(b)
(c)
(d)
2
Find: (i) P(X ≥ 3)
(ii) P(0 < X < 3)
(iii) P( X + 1 < 5)
Compute E(X).
Find the variance of X and hence the standard deviation of X.
Find the fraction of X that lies between (both end points inclusive) one standard deviation
of the mean.
8.
Suppose in a certain city, 40% of employed men drive to work. If 5 employed men are
selected at random,
what is the probability that exactly 2 drive to work?
what is the probability that at most 2 drive to work?
what is the probability that between 1 and 3, both inclusive, drive to work?
what are the mean and the standard deviation of X where X denotes the number of
employed men who drive to work?
(a)
(b)
(c)
(d)
Page 3 of 6
9.
(a)
(b)
(c)
10.
(a)
(c)
11.
(a)
(b)
12.
(a)
(b)
(c)
(d)
13.
(a)
(b)
(c)
(d)
(e)
The grade point averages of a large population of college students are approximately
normally distributed with a mean of 2.40 and a standard deviation of 0.80.
What fraction of the students will possess a grade point average in excess of 3.00?
If students possessing a grade point average equal to or less than 1.96 are dropped from
college, what percentage of the students will be dropped?
What percentage of the students will possess a grade point average between 2.00 and 3.20?
Suppose that the growth in inches during the tenth year of life of a North American boy is a
normal random variable with mean µ = 2 and standard deviation σ = 1. Find the probability
that a randomly selected North American boy in his tenth year will grow
more than 3 inches;
(b) less than 0.5 inch;
at least 1 inch;
(d) exactly 2 inches.
A random sample of size 81 is taken from a population having a mean of 128 and a
standard deviation of 6.3. The shape of the population distribution is skewed.
By the Central Limit Theorem:
(i) What is the distribution of X? (Give specific name). Is it exact or approximate?
(ii) What are the mean and the standard deviation of X?
What is the probability that X will:
(i) exceed 130.1?
(ii) fall below 126.6?
(iii) be exactly equal to 135.0?
A random sample of size 36 is taken from a normal population having mean of 20 and a
standard deviation of 4.5.
What is the distribution of X? (give specific name).
Is the distribution in (a) exact or approximate? Justify your answer.
What are the mean and the standard deviation of X?
What is the probability that:
(i) X will exceed 20.75?
(ii) X will fall below 19.80?
(iii) X will lie between 19.50 and 21.50?
Suppose in an algebra test, a random sample of 150 students has a mean score x = 78.1
with a standard deviation s = 9.5.
Give the point estimate of the population mean µ.
Compute the standard error of your estimate in (a).
Compute a 95% confidence interval for the population mean µ.
Explain the meaning of the confidence interval in (c).
Explain the meaning of the standard error in (b).
Page 4 of 6
14.
(a)
(b)
(c)
15.
(a)
(b)
(c)
A Gallup poll survey on the drinking habits of Americans estimated the percentage of
adults across the country that drink beer, wine, or hard liquor, at least occasionally. Of the
1500 adults interviewed, 975 said they drank.
Find a point estimate for the proportion, p, of Americans who drink.
Find the standard error of your estimate in (a).
Find 95% confidence interval for the population proportion, p.
An insurance company wants to estimate what proportion of its policyholders would be
interested in a $500 deductible on their collision coverage. The company randomly
sampled 1200 policyholders, and 780 said they would adopt this deductible if it were
available.
Find the point estimate for the population proportion p.
Find the standard error of your estimate in (a).
Find a 90% confidence interval for the population proportion p.
16.
The amount of sewage and industrial pollutants dumped into a body of water affects the
health of the water by reducing the amount of dissolved oxygen available for aquatic life.
Suppose that weekly readings are taken from the same location in a river over a two month
period (i.e. eighth observations are taken) and the following are obtained:
x = 4.95;
s = 0.45.
Use the above data to conduct a statistical test at α = 0.05 that the dissolved oxygen content
is less than 5.0 parts per million. (Hint: Use the 5-step procedure).
17.
A substitute teacher in a large high school district was unable to get full-time teaching
status because the district was not adding any more classes. The teacher claimed the
district was increasing class sizes above the stated average of 32.0 students. To test this
claim at α = 0.05, a student randomly selected 64 classes from the high schools in the
district and obtained x = 33.6 and s = 9.6.
State the hypotheses for this problem.
State the test statistic.
Find the rejection region and test the null hypothesis.
State an appropriate conclusion in terms of the original problem.
What type of error are you liable to in (c) and why?.
(a)
(b)
(c)
(d)
(e)
Page 5 of 6
18.
(a)
(b)
(c)
19.
The manager of a health maintenance organization claimed that the mean waiting time of
non-emergency patients does not exceed 30 minutes. To test this claim, the waiting times
for 25 patients were randomly selected on different days and the sample mean was found to
be 38.1 minutes with a standard deviation of 10 minutes.
Formulate the null and the alternative hypotheses in terms of the population mean.
State the test statistic and the rejection region corresponding to α = 0.025.
Obtain the value of the test statistic, make an appropriate decision and state your
conclusion.
Suppose that we have obtained independent samples from two normal populations with
means µ1 and µ2. The sample sizes, sample means, and sample standard deviations are as
(a)
(b)
follows:
x1 = 27.2
s1 = 5.9
n2 = 36
n1 = 40
Construct a 95% confidence interval for µ1 – µ2.
Compute a 99% confidence interval for µ1 – µ2.
(c)
(d)
Explain the meaning of the confidence interval in (b).
Based on your answer in (a), can the two populations means be equal? Why or why not?
20.
A paper company conducted an experiment to compare the mean time to unload shipments
of logs for two different unloading procedures. Two samples of 50 trucks each were
unloaded using method 1 and method 2. The sample means and standard deviations are
shown below in minutes.
method 1
method 2
n2 = 50
n1 = 50
x1 = 25.4
x2 = 27.3
s2 = 3.7
s1 = 3.1
x2 = 29.3
s2 = 4.8
Use the above data to conduct a statistical test at α = 0.05 that the mean time for method 1
is less than the mean time for method 2. (Hint: Use the 5-step procedure).
(a)
Many baseball experts consider pitching as the key to a successful season. A frequently
used measure of a pitching staff's effectiveness is its earned run average (ERA), which a
pitcher strives to keep as low as possible. The following summary statistics are on the
number of wins (y) and the earned run averages (x) of the 14 American League teams for
1990 season.
Σxi = 54.69 Σyi = 1133.0
Σ x i2 = 214.9087
Σ y i2 = 92815.0
Σxiyi = 4395.19
Find the least squares line ŷ = βˆ + βˆ x .
(b)
Give a brief interpretation of the values of βˆ0 and βˆ1 .
(c)
Calculate the correlation coefficient and interpret its value.
21.
0
1
Page 6 of 6
(a)
A chemist is interested in determining the weight loss y of a particular compound as a
function of the amount of time (x) the compound is exposed to the air. The simple linear
regression model is considered for a sample of size n = 12. The summary data are:
Σxi = 66.00 Σyi = 66.10
Σ x i2 = 378.00
Σ y i2 = 396.57
Σxiyi = 383.30
Find the least squares line ŷ = βˆ + βˆ x .
(b)
Give a brief interpretation of the values of βˆ0 and βˆ1 .
(c)
(d)
Predict the weight loss when the amount of time x = 5.
Calculate the correlation coefficient r and interpret its value. In what units is r expressed?
22.
0
Summary
Chapter in the text
2
3
4
5
6
7
9
1
Problem
1, 2, 3.
4, 5.
6, 7, 8, 9, 10, 11, 12.
13, 14, 15.
16, 17, 18.
19, 20.
21, 22.