268 Fundamentals of Statistics
Exercises
- True or false (and why): A 95% confi dence
interval that covers the range ( 2 5, 5) tells
you that the probability is 95% that m will
have a value between 2 5 and 5. - True or false (and why): Accepting the
null hypothesis means that the null
hypothesis is true. - True or false (and why): Rejecting the
null hypothesis means that the null
hypothesis is false. - Consider a sample of 25 normally distrib-
uted observations with a sample average
of 50.
a. Calculate the 95% confi dence interval
if s 520.
b. Calculate the 95% confi dence inter-
val if s is unknown but if the sample
standard deviation 5 20. - The nationwide mean price for a three-
year-old Honda Civic DX is $11,500 with
a known standard deviation of $600.
You check the newspaper and fi nd 9
three-year-old Civics in San Francisco
selling for an average price of $12,000.
You wonder whether the cost of Civics
in San Francisco is higher than for the
rest of the nation.
a. State your question about the price of
Civics in terms of a null and an alterna-
tive hypothesis. What are you assuming
about the distribution of Civic prices?
b. Will the alternative hypothesis be one
or two sided? Defend your answer.
c. Test your null hypothesis. Do you ac-
cept or reject it and at what p value?
Construct a 95% confi dence interval
for Civic prices in San Francisco.
d. Redo your analysis, but this time as-
sume that the sample size is 10 with
a sample average of $12,000 and a
sample standard deviation of $600.
Assume that you don’t know the
value of the nationwide standard
deviation.- In tests of stereo speakers, ten American-
made speakers had an average perform-
ance rating of 90 with a standard deviation
of 5. Five imported speakers had an aver-
age rating of 85 with a standard devia-
tion of 4.
a. Write a null and an alternative hy-
pothesis comparing the two types of
speakers.
b. Test the null hypothesis. What is the
p value?
c. If you decide to change the signifi -
cance level to 10%, does your conclu-
sion change? - Derive the formula for the t confi dence
interval based on the defi nition of the
t statistic shown earlier in this chapter. - You want to continue to study the nurs-
ing home data discussed in this chapter.
Explore whether there is a signifi cant
difference between rural and nonrural
homes in terms of size (as expressed by
the number of beds in the homes).
a. Open the Nursing Home workbook
from the Chapter06 folder. Save the
workbook as Nursing Home Beds to
the same folder.
b. Write down a set of hypotheses for
exploring the question of whether the
numbers of beds in rural and nonrural
homes differ.
c. Apply a two-sample t test to the data.
Report your results assuming a pooled
estimate of the standard deviation and
assuming an unpooled estimate. What
are the p value and confi dence inter-
val under each assumption?