Section 6.1 Solutions
6.5
Since the sample size is large (n
= 31), we know the distribution of 
is approximately
normal (by the Central Limit Theorem), so we can create the confidence interval
even though we don’t know if the population is normal.
For 95% confidence, 
= 1.96 (from reverse look-up of the area 0.975 in the normal
table). Based on the Empirical Rule, you could also use = 2. Then the margin of error is 
U/l, and the 95% confidence interval for the mean serum TRAP
for young women is 
(10.91 U/l, 15.49 U/l).
6.7
For 99% confidence, = 2.575 (from reverse
look-up of the area 0.995 in the normal table). Note: if you use magical Table
D, = 2.576. Then the 99%
confidence interval for the mean serum TRAP in young women is
= (10.19 U/l, 16.21 U/l). This confidence interval is wider than the
interval given in problem 6.5 (the tradeoff for more confidence is a wider
interval).
6.9
Recall the margin of error is 
. In both scenarios, = 1.96 and n = 100, so the only difference in the
margin of error for the two scenarios is the population standard deviation of
heights. In scenario B, this standard deviation will be smaller, since there
will be less variability in heights when considering only third-graders (there
will be more variability when considering the heights of kindergartners through
fifth-graders combined). Hence, scenario B will have a smaller margin of error.
6.13
The sample size is 25. Even though it doesn’t meet our quick check (
), it’s probably large enough, unless the population of study
times is severely skewed. Hence, we know the distribution of is approximately
normal (by the Central Limit Theorem), so we can create the confidence interval
even though we don’t know if the population is normal.
a.
For 95%
confidence, = 1.96 (from reverse look-up of the area 0.975 in the normal
table). Based on the Empirical Rule, you could also use = 2. Then the 95% confidence interval for the mean study time
of all statistics students is 
= (66.28 minutes, 93.72
minutes).
b. The
confidence interval provides estimates of the average study time not of an individual study time. Hence, we cannot say that 95% of the students
studied between 66.28 minutes and 93.72 minutes.
6.14
The linear transformation to convert from minutes to hours is 
.
a. Then the mean in hours is 80/60 = 1.33, and
the standard deviation in hours is 35/60 = 0.583.
b.
The 95%
confidence interval for the mean study time (in hours) of all statistics
students is 
= (1.10 hours, 1.56
hours).
c. We could have simply converted the confidence interval endpoints in the previous problem to hours: (66.28/60 = 1.10 hours, 93.72/60 = 1.56 hours).
6.19
Since the sample size is large (n
= 44), we know the distribution of is approximately
normal (by the Central Limit Theorem), so we can create the confidence interval
even though we don’t know if the population is normal.
For 95% confidence, = 1.96 (from reverse look-up of the area 0.975 in the normal
table). Based on the Empirical Rule, you could also use = 2. From the sample data, 
. Then the 95% confidence interval is 
= (31.84, 38.34). Hence, we are 95% confident the population mean DRP score is between 31.84 and 38.34. Our
confidence is in the method we use to create the interval—if the sampling were
(hypothetically) done repeatedly, then 95% of the intervals created would
contain the true population mean.
6.22
The form of the confidence interval is 
, so the margin of error is 
.
For 95% confidence, = 1.96 (from reverse look-up of the area 0.975 in the normal
table). Based on the Empirical Rule, you could also use = 2.
Then we need to find n such
that 
Since we always round up, we should take a sample of size 1945.
6.23
A larger margin of error should allow for a smaller sample size.
Then we need to find n such
that 
Since we always round up, we should
take a sample of size 487. This value is obviously much smaller than what we needed
in problem 6.22.
6.25
The form of the confidence interval is , so the margin of error is .
For 95% confidence, = 1.96 (from reverse look-up of the area 0.975 in the normal
table). Based on the Empirical Rule, you could also use = 2.
Then we need to find n such
that 
Since we always round up, we should take a sample of size 41.
6.26
In similar previous studies, about 20% of the subjects drop out before
the study is completed. If n is the
sample size, this means than 0.8n of
the subjects complete the study. We want to find n, such that 0.8n meets
the requirements of problem 6.25:
Since we always round up, we should take a sample of size 51.
6.31
a. The value of 37% is the result from one sample. If we took another sample, then we would probably get a different value. That is, there is variability in the sampling process. The only way to unequivocally find the true percentage is to take an accurate census. We can reasonably use 37% as an estimate of the population percentage, but we must realize that there is some variability associated with this estimate.
b. The confidence we have is in the method we use to create the interval. If the sampling were (hypothetically) done repeatedly, then 95% of the intervals created would contain the true percentage of all Americans who say football is their favorite sport to watch on television. Put another way, we are using a method that gives correct results 95% of the time (where a “correct result” is an interval that contains the true population percentage).
c.
Confidence intervals typically take on the following
form: estimate 
margin of error.
Hence, the 95% confidence interval is 37 3 = (34%, 40%).
d. A poll conducted in June would probably produce different results, since fans may change their responses based on what they are currently watching.