# Confidence interval

```Confidence interval
1. A sample of 49 observations is taken from a normal population. The sample mean is 55, and the
sample standard deviation is 10. Determine the 99 percent confidence interval for the population mean.
15. The owner of the West End Kwick Fill Gas Station wished to determine the proportion of customers
who use a credit card or debit card to pay at the pump. He surveys 100 customers and finds that 80 paid
at the pump. a. Estimate the value of the population proportion. b. Compute the standard error of the
proportion. c. Develop a 95 percent confidence interval for the population proportion. d. Interpret your
findings.
x̅ &plusmn; z
α σ
2 √n
Single population mean [z-test]
5. The manufacturer of the X-15 steel-belted radial truck tire claims that the mean mileage the tire can
be driven before the tread wears out is 60,000 miles. The Crosset Truck Company bought 48 tires and
found that the mean mileage for their trucks is 59,500 miles with a standard deviation of 5,000 miles. Is
Grosset's experience different from that claimed by the manufacturer at the .05 significance level?
𝑍𝑐 =
̅−&micro;
𝒙
𝝈
√𝒏
Single population mean [t-test]
17. The Rocky Mountain district sales manager of Rath Publishing, Inc., a college textbook publishing
company, claims that the sales representatives make an average of 40 sales calls per week on
professors. Several reps say that this estimate is too low. To investigate, a random sample of 28 sales
representatives reveals that the mean number of calls made last week was 42. The standard deviation of
the sample is 2.1 calls. Using the .05 significance level, can we conclude that the mean number of calls
per salesperson per week is more than 40?
𝒕𝑐 =
̅−&micro;
𝒙
𝒔
√𝒏
Df = n-1
Single population proportion [z-test]
11. The National Safety Council reported that 52 percent of American turnpike drivers are men. A
sample of 300 cars traveling southbound on the New Jersey Turnpike yesterday revealed that 170 were
driven by men. At the .01 significance level, can we conclude that a larger proportion of men were
driving on the New Jersey Turnpike than the national statistics indicate?
𝒁𝒄 =
̅ − 𝒏𝒑
𝒙
√𝒏𝒑𝒒
Two population mean [z-test]
3. The Gibbs Baby Food Company wishes to compare the weight gain of infants using their brand versus
their competitor's. A sample of 40 babies using the Gibbs products revealed a mean weight gain of 7.6
pounds in the first three months after birth. The standard deviation of the sample was 2.3 pounds. A
sample of 55 babies using the competitor's brand revealed a mean increase in weight of 8.1 pounds,
with a standard deviation of 2.9 pounds. At the .05 significance level, can we conclude that babies using
the Gibbs brand gained less weight? Compute the p-value and interpret it.
𝒛𝒄 =
̅𝟏 − 𝒙
̅𝟐
𝒙
𝟐
𝟐
√ 𝑺𝟏 + 𝑺𝟐
𝒏𝟏 𝒏𝟐
Two population mean [t-test]
35. The Willow Run Outlet Mall has two Gap Outlet Stores, one located on Peach Street and the other
on Plum Street. The two stores are laid out differently, but both store managers claim their layout
maximizes the amounts customers will purchase on impulse. A sample of 10 customers at the Peach
Street store revealed they spent the following amounts more than planned: \$17.58, \$19.73, \$12.61,
\$17.79, \$16.22, \$15.82, \$15.40, \$15.86, \$11.82, and \$15.85. A sample of 14 customers at the Plum
Street store revealed they spent the following amounts more than they planned: \$18.19, \$20.22,
\$17.38, \$17.96, \$23.92, \$15.87, \$16.47, \$15.96, \$16.79, \$16.74, \$21.40, \$20.57, \$19.79, and \$14.83. At
the .01 significance level, is there a difference in the mean amounts purchased on impulse at the two
stores?
𝒕𝒄 =
̅𝟏 − 𝒙
̅𝟐
𝒙
√𝒔𝟐 𝒑(
𝟏
𝟏
𝒏𝟏 + 𝒏𝟐 )
(𝒏𝟏 − 𝟏)𝑺𝟐𝟏 + (𝒏𝟐 − 𝟏)𝑺𝟐𝟐
𝒔 𝒑=
𝒏𝟏 + 𝒏𝟐 − 𝟐
𝟐
𝒅𝒇 = 𝒏𝟏 + 𝒏𝟐 − 𝟐
Two population proportion [t-test]
10. The Roper Organization conducted identical surveys in 1995 and 2005. One question asked women
was &quot;Are most men basically kind, gentle, and thoughtful?&quot; The 1995 survey revealed that, of the 3,000
women surveyed, 2,010 said that they were. In 2005, 1,530 of the 3,000 women surveyed thought that
men were kind, gentle, and thoughtful. At the .05 level, can we conclude that women think men are less
kind, gentle, and thoughtful in 2005 compared with 1995?
𝒁𝒄 =
𝑷𝟏 − 𝑷𝟐
𝟏
𝟏
𝑷𝒒(𝒏 + 𝒏 )
𝟏
𝟐
𝑷𝟏 =
𝑿𝟏
𝒏𝟏
𝑷𝟐 =
𝑿𝟐
𝒏𝟐
𝑷=
𝑿𝟏 + 𝑿𝟐
𝒏𝟏 + 𝒏𝟐
Co-relation
9. The Pennsylvania Refining Company is studying the relationship between the pump price of gasoline
and the number of gallons sold. For a sample of 20 stations last Tuesday, the correlation was .78. At the
.01 significance level, is the correlation in the population greater than zero?
𝒕𝒄 =
𝒓√𝒏 − 𝟐
√𝟏 − 𝒓𝟐
Df = n-2
Goodness-of-Fit Test:
18. The publisher of a sports magazine plans to offer new subscribers one of three gifts: a sweatshirt
with the logo of their favorite team, a coffee cup with the logo of their favorite team, or a pair of
earrings also with the logo of their favorite team. In a sample of 500 new subscribers, the number
selecting each gift is reported below. At the .05 significance level, is there a preference for the gifts or
should we conclude that the gifts are equally well liked?
Frequency
Sweatshirt
Coffee cup
183
175
Earrings
Df = k-p-1
𝝀𝟐 = {
[𝒐𝒊 − 𝒆𝒊 ]
𝒆𝒊
𝒐𝒊 = 𝐅𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲
𝒆𝒊 = 𝒏𝒑𝒊
142
```