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STA 2023 - Quiz 3 (Answer Key)
Spring 2003
Quiz proceedings:
 No cheating
 Show all relevant work to receive full credit – numbers in parentheses are point values for each problem
 Please turn off your cellular telephone
 Not permitted for use during the quiz: harmonicas, Reebok Pump shoes, canned tuna, troll dolls
 Good luck!
1. The lengths of smallmouth bass in Wisconsin are known have an average length of 210 mm with a
standard deviation of 30 mm. Consider random samples of size 36. 6.29
a. Calculate the mean of the distribution of the sample mean. (2)
 x    210
b. Calculate the standard deviation of the distribution of the sample mean. (3)

30
30
x 


5
6
n
36
c. Draw a graph of the distribution of the sample mean. (5)
195
200
205
210
215
220
225
d. What is the probability that you observe a sample mean greater than 230 mm? (5)
230  210 

P( x  230)  P z 
  P( z  4)  0
5


e. What is the probability that you observe a sample mean less than 200 mm? (5)
200  210 

P( x  200)  P z 
  P( z  2)  .0228 (from Table IV)
5


f. What is the probability that you observe sample mean between 205 and 215 mm? (5)
215  210 
 205  210
P(205  x  215)  P
z
  P(1  z  1)  2  .3413  .6826 (.3413
5
5


from Table IV)
2. We are interested in doing a study on undergraduate Psychology majors at UCF and we wish to find
the average GPA of all current undergraduate Psychology majors. In a random sample of 81
students, we found an average GPA of 2.70 with a standard deviation of .36. Assume that the
original population is normally distributed. 7.19
a. Find a point estimate for the true mean GPA of undergraduate Psychology majors. (5)
x  2.70
b. Calculate a 95% confidence interval for the true mean GPA of undergraduate Psychology
majors. (5)

.36
x  z 2
 2.70  1.96
 2.70  .0784  (2.62,2.78)  2.62 <  < 2.78
n
81
c. Interpret the interval found in part b. (5)
We are 95% confident that the true average GPA of Psychology majors at UCF is
between 2.6216 and 2.7784.
d. Suppose that we wish construct a 95% confidence interval using the standard deviation found
in our original sample. How many students would we have to sample in order to have our
interval accurate to within .02? (5)
( z 2 ) 2  2 (1.96) 2 (.36) 2
n

 1244.6784  1245
B2
(.02) 2
3. A random sample of 25 U.S. adult males who jogs at least 15 miles per week had a mean pulse rate
of 52.6 beats per minute and a standard deviation of 3.2 beats per minute. Assume the population is
normally distributed. 7.76
a. Find a 99% confidence interval for the mean pulse rate of all U.S. adult males who jog at
least 15 miles per week. (5)
s
3.2
x  t v , 2
 52.6  2.797
 52.6  1.79008  (50.81,54.39)  50.81 <  < 54.39
n
25
b. Interpret the interval found in part a. (5)
We are 99% confident that the true average pulse rate of U.S. males who jog at least 15
miles per week is between 50.81 and 54.39 beats per minute.
c. If the mean pulse rate for all U.S. adult males is 72 beats per minute, does it appear that
jogging at least 15 miles per week reduces the mean pulse rate for adult males? Explain. (5)
Since the interval does not include the national average of 72 we can conclude that there
is a significant difference between jogging at least 15 miles per week and not jogging.
Since the interval lies entirely below the national average, it does appear that jogging at
least 15 miles per week reduces the mean pulse rate.
4. Suppose we wish to conduct a survey for the true proportion of Americans who will vote for George
Bush, should he seek reelection. In a random sample of 10,000 people, we find that 4,120 people
say they will vote for President Bush in the upcoming election, should he seek reelection. 7.43
a. Find a point estimate for the true proportion of Americans who will vote for President Bush,
should he seek reelection. (5)
x 4,120
pˆ  
 .412
n 10,000
b. Will constructing a confidence interval here be appropriate? Explain. (5)
pˆ qˆ
.412  .588
pˆ  3
 .412  3
 .412  .0147  (.3973,.4267)  (0,1)
n
10,000
Yes, constructing a confidence interval here would be appropriate.
c. Regardless of your answer in part b, construct a 99% confidence interval for the true
proportion of Americans who will vote for President Bush, should he seek reelection. (5)
pˆ qˆ
.412  .588
pˆ  z 2
 .412  2.575
 .412  .0127  (.3993,.4247)  .3993 < p < .4247
n
10,000
d. Interpret the interval found in part c. (5)
We are 99% confident that the true proportion of Americans who will vote for George
Bush should he seek reelection is between .3993 and .4247.
e. Suppose we wish to construct a 99% confidence interval using the voting results from the
2000 election. If we want to be accurate to within 3% of the true proportion, how many
people should be included in our sample? In the 2000 election, President Bush received
approximately 47.9% of the popular vote. (5)
( z 2 ) 2 pq (2.575) 2 (.479)(.521)
n

 1838.59  1839
B2
(.03) 2
5. In the construction of confidence intervals, sometimes our computed intervals are not as precise as
we would like. In order to narrow our interval, list the two things that we talked about in class that
can make our interval more precise. (5) 3/4 class notes
a. Increase sample size
b. Decrease confidence
6. We wish to construct a confidence interval for . Given the following original populations, sample
sizes, and levels of confidence, state whether the critical value is a value of z or t, and calculate the
critical value in question. Assume that the standard deviation of the population is unknown (it
usually is). Do not try to calculate the entire interval. If you are forced to result to the methods of
nonparametric statistics of Chapter 14, simply write “Chapter 14”. 7.1, 7.22
a. Normal population, sample size of 37, 98% confidence (3)
z 2  z.02 2  z.01  2.33
b. Normal population, sample size of 7, 98% confidence (3)
t v , 2  t 6,.02 2  t 6,.01  3.143
c. Beta population, sample size of 7, 98% confidence (3)
Chapter 14 (since n < 30 and population is not normal)
7. What’s your favorite movie? (1)
~ The End ~