Chapter 12—Estimation: Comparing two populations
MULTIPLE CHOICE
1. If two random samples of sizes
1
n
and
2
n
are selected independently from two populations with
means
1
and
2
, then the mean of the sampling distribution of the sample mean difference,
21 XX −
, equals:
A.
1
+
2
.
B.
1
–
2
.
C.
1
/
2
.
D.
1
2
.
2. If two random samples of sizes
1
n
and
2
n
are selected independently from two non-normally
distributed populations, then the sampling distribution of the sample mean difference,
21 XX −
, is:
A.
always non-normal.
B.
always normal.
C.
approximately normal only if
1
n
and
2
n
are both larger than 30.
D.
approximately normal regardless of
1
n
and
2
n
.
3. If two populations are normally distributed, the sampling distribution of the sample mean difference,
21 XX −
, will be:
A.
normally distributed only if both population sizes are greater than 30.
B.
normally distributed.
C.
normally distributed only if at least one of the sample sizes is greater than 30.
D.
approximately normally distributed.
4. Two samples are selected at random from two independent normally distributed populations. Sample 1
has 49 observations and has a mean of 10 and a standard deviation of 5. Sample 2 has 36 observations
and has a mean of 12 and a standard deviation of 3. The standard error of the sampling distribution of
the sample mean difference,
21 XX −
, is:
A.
0.1853.
B.
0.7602.
C.
0.7331.
D.
0.8719.
5. If two random samples of sizes
1
n
and
2
n
are selected independently from two populations with
variances
2
1
and
2
2
, then the standard error of the sampling distribution of the sample mean
difference,
21 XX −
, equals:
A.
21
2
2
2
1/)( nn
−
.
B.
21
2
2
2
1/)( nn
+
.
C.
2
2
2
1
2
1
nn
−
.
D.
2
2
2
1
2
1
nn
+
.
6. The expected value of the difference of two sample means equals the difference of the corresponding
population means:
A.
only if the populations are normally distributed.
B.
only if the samples are independent.
C.
only if the populations are approximately normal and the sample sizes are large.
D.
the statement is correct under all circumstances.
7. In constructing a confidence interval estimate for the difference between two population proportions,
we:
A.
pool the population proportions when the populations are normally distributed.
B.
pool the population proportions when the population means are equal.
C.
pool the population proportions when they are equal.
D.
never pool the population proportions.
8. In constructing a 99% confidence interval estimate for the difference between the means of two
normally distributed populations, where the unknown population variances are assumed not to be
equal, summary statistics computed from two independent samples are as follows:
n1 = 28
x-bar1 = 123
s1 = 8.5
n2 = 45
x-bar2 = 105
s2 = 12.4
The lower confidence limit is:
A.
24.485.
B.
11.515.
C.
13.116.
D.
22.884.
9. In constructing a confidence interval estimate for the difference between the means of two normally
distributed populations, using two independent samples, we:
A.
pool the sample variances when the unknown population variances are equal.
B.
pool the sample variances when the population variances are known and equal.
C.
pool the sample variances when the population means are equal.
D.
never pool the sample variances.
10. The expected value of the difference of the two sample means equals the difference of the
corresponding population means:
A.
only if the populations are normally distributed.
B.
only if the samples are independent.
C.
only if the populations are approximately normal and the sample sizes are large.
D.
The statement is correct under all circumstances.
SHORT ANSWER
1. Two independent random samples of 25 observations each are drawn from two normal populations.
The parameters of these populations are:
Population 1:
µ = 150, = 50.
Population 2:
µ = 130, = 45.
Find the probability that the mean of sample 1 will exceed the mean of sample 2.
2. Two independent random samples are drawn from two normal populations. The sample sizes are 20
and 25, respectively. The parameters of these populations are:
Population 1:
µ = 505, = 10.
Population 2:
µ = 4750, = 7.
Find the probability that the difference between the two sample means (X1-bar – X2–bar) is between 25
and 35.
3. Suppose that the starting salaries of male workers are normally distributed with a mean of $56 000 and
a standard deviation of $12 000. The starting salaries of female workers are normally distributed with a
mean of $50 000 and a standard deviation of $10 000. A random sample of 50 male workers and a
random sample of 40 female workers are selected.
a. What is the sampling distribution of the sample mean difference – ? Explain.
b. Find the expected value and the standard error of the sample mean difference.
c. What is the probability that the sample mean salary of female workers will not exceed that of the
male workers?
4. Suppose that the starting salaries of finance graduates from university A are normally distributed with
a mean of $36 750 and a standard deviation of $5320. The starting salaries of finance graduates from
university B are normally distributed with a mean of $34 625 and a standard deviation of $6540. If
simple random samples of 50 finance graduates are selected from each university, what is the
probability that the sample mean of university A graduates will exceed that of university B graduates?
TRUE/FALSE
1. If two random samples, each of size 36, are selected independently from two populations with
variances of 42 and 50, then the standard error of the sampling distribution of the sample mean
difference,
21 XX −
, equals 2.5556.
2. If two random samples of sizes 30 and 45 are selected independently from two non-normal populations
with means of 53 and 57, then the mean of the sampling distribution of the sample mean difference,
21 XX −
, equals -4.