Stat 281: Intro to Statistics
F2004, Dr. Galster
Test 3 Practice
Name___KEY______
1. The expected value of a sample mean is
a. smaller than the population mean.
b. equal to the population variance.
c. equal to the population mean.
d. different for every sample taken.
2. The symbol commonly used for a sample mean is
a. x .
b. s .
c. sx .
d.  .
3. The Central Limit Theorem says that
a. the variance of x decreases as n increases.
b. the distribution of sample means is approximately normal if n is large enough.
c. the normal distribution may be used to estimate x .
d. the sample mean of a normal distribution is normal.
4. The sample mean is a random variable because
a. its value changes depending on the population mean.
b. it is a function of other random variables.
c. you never know what its value will be.
d. it has a mean and variance.
5. The variance of a sample mean depends on
a. the variance of the population only.
b. the sample size and the population mean.
c. the variance of the population and the population mean.
d. the variance of the population and the sample size.
6. The effect of increasing the sample size is to
a. decrease the variability of the sample mean.
b. decrease the variance of the population..
c. increase the confidence level.
d. increase the standard deviation of the sample mean.
7. The confidence level of an interval estimate is
a. the probability that μ is in the interval.
b. the percent of sample means that are in the interval.
c. the percent of such intervals that would contain the parameter in the long run.
d. the amount of confidence we feel in our estimation procedure.
8. Find the following probabilities from a standard normal distribution.
a) P(Z>2.08)=.0188
b) P(Z<1.05)=.8531
c) P(-2.06<Z<1.06)=.8357
d) P(0<Z<3)=.4987
9. Find the following probabilities from a normal distribution with μ=100 and σ=5.
a) P(X>112.5)=.0062
b) P(X<87.4)=.0059
c) P(97.1<X<105.4)=.5789
d) P(X<112.5)=.9938
10. Find a z-score associated with each probability.
a) P(0<Z<z)=.4382 z=1.54
b) P(Z<z)=.0011 z= -3.06
c) P(-z<Z<z)=.4108 z=.54
d) z(.0055)=2.54
11. Find a t-value with the given degrees of freedom and upper tail probability.
a) df=14, α=.10 t(14,.10)=1.345
b) df=23, α=.01 t(23,.01)=2.500
c) df=40, α=.05 t(40,.05)=1.684
d) df=7, α=.025 t(7,.025)=2.365
12. Give the table value that should be used to construct a confidence interval for μ with:
a) Population normal, n=15, σ=9, s=8.4, α=.10 z(.05)=1.645
b) n=61, s=15, α=.05
c) n=100, σ=95, 95% CI
t(60,.025)=2.000 (or possibly z(.025)=1.96)
z(.025)=1.96
d) Population normal, n=10, s=6, 98% CI t(9,.01)=2.821
13. Let X be a random variable representing the weight of a fish caught using a net on a
large fishing boat. If the mean weight of all fish caught in this manner is 5 kg and the
standard deviation is 1.2 kg, describe the distribution of the sample mean, if the
sample is one batch of 100 fish caught.
It would be approximately normal, with mean 5 kg and standard error .12 kg.
14. The diameters of apples in an orchard are normally distributed with a mean of 2.5
inches and a standard deviation of 0.50 inches.
a. What is the probability that a randomly selected apple will have a diameter of more
than 3 in.?
.1587
b. What is the probability that, in a sample of 25 apples, the sample mean will be
more than 2.7 in.?
.0228
c. What is the probability that the sample mean (n=25) will fall between 2.4 and 2.6?
.6826
d. Suppose a confidence interval is built from a sample of 25 apples, and the result is
(2.204, 2.596). What sample mean would have produced this result, and what
confidence level would have been used?
sample mean 2.4, confidence level = 95%
15. Ten randomly selected shut-ins were each asked to list how many hours of television
they watched per week. The results are 82, 66, 90, 84, 75, 88, 80, 94, 110, and 91.
Determine the 90% confidence interval for the mean number of hours of television
watched per week. Assume the number of hours is normally distributed and the
population standard deviation is known to be 12.
86  1.645(12 / 10)
(79.75, 92.24)
b) Redo the confidence interval assuming the standard deviation is unknown.
s=11.84 t(9,.05)=1.823
86  1.833(3.7446)
(79.14,92.86)
c) What sample size would be required to estimate the mean number of hours in a),
above, with a bound of one hour?
390
16. A poll was conducted to determine the level of support for Candidate X. 400 people
were asked if they would vote for him, and 160 said yes. Construct a 95% CI for the
proportion of people who favor Candidate X.
(.4)(.6)
400
.4  1.96
(.352,.448)
17. Another poll is planned to follow up on the one in the previous question. This time
the desired margin of error is 4%. What sample size should be used for 95%
confidence?
577
x 

n
x  z / 2 x
sx 
s
n
x  t ( df , / 2) s x
pˆ  z / 2
pˆ (1  pˆ )
n