Download 31. A new weight-watching company, Weight Reducers International

31. A new weight-watching company, Weight Reducers International, advertises that those who join
will lose, on the average, 10 pounds the first two weeks with a standard deviation of 2.8 pounds. A
random sample of 50 people who joined the new weight reduction program revealed the mean loss to
be 9 pounds. At the .05 level of significance, can we conclude that those joining Weight Reducers on
average will lose less than 10 pounds? Determine the p-value.
𝐻𝑜: 𝜇 ≥ 10
𝐻𝑎: 𝜇 < 10
9−10
Test statistic: 𝑧 = 2.8/√50 = −2.525
Critical value: 𝑧𝑐𝑟 = −1.645
Reject the null hypothesis. We have sufficient evidence to conclude that those joining Weight
Reducers on average will lose less than 10 pounds.
p-value = 0.0058
32. Dole Pineapple, Inc., is concerned that the 16-ounce can of sliced pineapple is being overfilled.
Assume the standard deviation of the process is .03 ounces. The qualitycontrol department took a
random sample of 50 cans and found that the arithmetic mean weight was 16.05 ounces. At the 5
percent level of significance, can we conclude that the mean weight is greater than 16 ounces?
Determine the p-value.
𝐻𝑜: 𝜇 ≤ 16
𝐻𝑎: 𝜇 > 16
Test statistic: 𝑧 =
16.05−16
0.03/√50
= 11.785
Critical value: 𝑧𝑐𝑟 = 1.645
Reject the null hypothesis. We have sufficient evidence to conclude that the mean weight is greater
than 16 ounces.
p-value = 0.0000
38. A recent article in The Wall Street Journal reported that the 30-year mortgage rate is now less
than 6 percent. A sample of eight small banks in the Midwest revealed the following 30-year rates (in
percent):
4.8
5.3
6.5
4.8
6.1
5.8
6.2
5.6
At the .01 significance level, can we conclude that the 30-year mortgage rate for small banks is less
than 6 percent? Estimate the p-value.
𝐻𝑜: 𝜇 ≥ 6
𝐻𝑎: 𝜇 < 6
𝑥̅ = 5.638
𝑠 = 0.635
5.638−6
Test statistic: 𝑡 = 0.635/√8 = −1.616
degree of freedom = 8 – 1 = 7
Critical value: 𝑡𝑐𝑟 = −2.998
We cannot reject the null hypothesis. We don’t have sufficient evidence to conclude that the 30-year
mortgage rate for small banks is less than 6 percent.
p-value = 0.0751
27. A recent study focused on the number of times men and women who live alone buy take-out
dinner in a month. The information is summarized below.
Statistic
Sample mean
Population standard deviation
Sample size
Men
24.51
4.48
35
Women
22.69
3.86
40
At the .01 significance level, is there a difference in the mean number of times men and women order
take-out dinners in a month? What is the p-value?
𝐻𝑜: 𝜇1 − 𝜇2 = 0
𝐻𝑎: 𝜇1 − 𝜇2 ≠ 0
Test statistic: 𝑧 =
24.51−22.69
2
2
√4.48 +3.86
35
= 1.871
40
Critical value: 𝑧𝑐𝑟 = 2.326
We cannot reject the null hypothesis. We don’t have sufficient evidence to conclude that there is a
difference in the mean number of times men and women order take-out dinners in a month.
p-value = 0.031
46. Grand Strand Family Medical Center is specifically set up to treat minor medical emergencies for
visitors to the Myrtle Beach area. There are two facilities, one in the Little River Area and the other in
Murrells Inlet. The Quality Assurance Department wishes to compare the mean waiting time for
patients at the two locations. Samples of the waiting times, reported in minutes, follow:
Location
Waiting Time
Little River
31.73 28.77 29.53 22.08 29.47 18.60 32.94 25.18 29.82 26.49
Murrells Inlet 22.93 23.92 26.92 27.20 26.44 25.62 30.61 29.44 23.09 23.10 26.69 22.31
Assume the population standard deviations are not the same. At the .05 significance level, is there a
difference in the mean waiting time?
𝐻𝑜: 𝜇1 − 𝜇2 = 0
𝐻𝑎: 𝜇1 − 𝜇2 ≠ 0
𝑥̅1 = 27.461
𝑠1 = 4.440
𝑛1 = 10
𝑥̅2 = 25.689
𝑠2 = 2.685
𝑛2 = 12
Test statistic: 𝑡 =
27.461−25.689
2
√4.440 +2.685
10
2
= 1.105
12
degree of freedom = 14
Critical value: 𝑡𝑐𝑟 = ∓2.145
We cannot reject the null hypothesis. We don’t have sufficient evidence to conclude that there is a
difference in the mean waiting times.
52. The president of the American Insurance Institute wants to compare the yearly costs of
auto insurance offered by two leading companies. He selects a sample of 15 families,
some with only a single insured driver, others with several teenage drivers, and pays each
family a stipend to contact the two companies and ask for a price quote. To make the
data comparable, certain features, such as the deductible amount and limits of liability, are
standardized. The sample information is reported below. At the .10 significance level,
can we conclude that there is a difference in the amounts quoted?
Family
Becker
Berry
Cobb
Progressive
Car Insurance
$2,090
1,683
1,402
GEICO
Mutual Insurance
$1,610
1,247
2,327
Debuck
DuBrul
Eckroate
German
Glasson
King
Kucic
Meredith
Obeid
Price
Phillips
Tresize
1,830
930
697
1,741
1,129
1,018
1,881
1,571
874
1,579
1,577
860
1,367
1,461
1,789
1,621
1,914
1,956
1,772
1,375
1,527
1,767
1,636
1,188
𝐻𝑜: 𝜇𝐷 = 0
𝐻𝑎: 𝜇𝐷 ≠ 0
𝑥̅𝐷 = 246.33
𝑠𝐷 = 546.96
246.33−0
Test statistic: 𝑡 = 546.96/√15 = 1.744
Degree of freedom = 15 – 1 = 14
Critical value: 𝑡𝑐𝑟 = 1.761
We cannot reject the null hypothesis. We don’t have sufficient evidence to conclude that there is a
difference in the amounts quoted.
23. A real estate agent in the coastal area of Georgia wants to compare the variation in the selling
price of homes on the oceanfront with those one to three blocks from the ocean. A sample of 21
oceanfront homes sold within the last year revealed the standard deviation of the selling prices was
$45,600. A sample of 18 homes, also sold within the last year, that were one to three blocks from the
ocean revealed that the standard deviation was $21,330. At the .01 significance level, can we
conclude that there is more variation in the selling prices of the oceanfront homes?
𝐻𝑜: 𝜎12 ≤ 𝜎22
𝐻𝑎: 𝜎12 > 𝜎22
45,6002
Test statistic: 𝐹 = 21,3302 = 4.57
degree of freedom numerator = 21 – 1 = 20
degree of freedom denominator 18 – 1 = 17
Critical value: 𝐹𝑐𝑟 = 3.607
We reject the null hypothesis. We have sufficient evidence to conclude that there is more variation in
the selling prices of the oceanfront homes.
28. The following is a partial ANOVA table.
Sum of Mean
Squares
320
280
500
Source
Treatment
Error
Total
df
2
9
11
Square
160
20
F
8
Complete the table and answer the following questions. Use the .05 significance level.
a. How many treatments are there?
b. What is the total sample size?
c. What is the critical value of F?
d. Write out the null and alternate hypotheses.
e. What is your conclusion regarding the null hypothesis?
a. 3
b. 12
c. 4.256
d.
Ho: μ1 = μ2 = μ3
Ha: at least one of the treatment means is different
e. Reject the null hypothesis.
19. In a particular market there are three commercial television stations, each with its own evening
news program from 6:00 to 6:30 P.M. According to a report in this morning’s local newspaper, a
random sample of 150 viewers last night revealed 53 watched the news on WNAE (channel 5), 64
watched on WRRN (channel 11), and 33 on WSPD (channel 13). At the .05 significance level, is there
a difference in the proportion of viewers watching the three channels?
Ho: There is no difference
Ha: There is difference
Expected number of viewers =
Test statistic: 𝑋 2 =
(53−50)2
50
+
150
3
= 50
(64−50)2
50
+
(33−50)2
50
= 9.88
Degree of freedom = 3 – 1 = 2
2
Critical value: 𝑋𝑐𝑟
= 5.991
We reject the null hypothesis. We have sufficient evidence to conclude that there is a difference in the
proportion of viewers watching the three channels.
20. There are four entrances to the Government Center Building in downtown Philadelphia.
The building maintenance supervisor would like to know if the entrances are equally utilized.
To investigate, 400 people were observed entering the building. The number using each entrance is
reported below. At the .01 significance level, is there a difference in the use of the four entrances?
Entrance
Main Street
Broad Street
Cherry Street
Walnut Street
Total
Frequency
140
120
90
50
400
Ho: There is no difference
Ha: There is difference
Expected number of viewers =
Test statistic: 𝑋 2 =
(140−100)2
100
+
400
4
= 100
(120−100)2
100
+
(90−100)2
100
+
(50−100)2
100
= 46
Degree of freedom = 4 – 1 = 3
2
Critical value: 𝑋𝑐𝑟
= 11.34
We reject the null hypothesis. We have sufficient evidence to conclude that there is a difference in the
use of the four entrances.