Stat 2470, Homework #8, Fall 2014
Name ______________________________________
Instructions: Show work or give calculator commands used to solve each problem. You may use Excel or
other software for any graphs. Be sure to answer all parts of each problem as completely as possible,
and attach work to this cover sheet with a staple.
1. The National Health Statistics Reports dates Oct. 22, 2008 included the following information on
the heights in inches for non-Hispanic white females.
Age
Sample Size
Sample Mean
Standard Error Mean
20-39
866
64.9
0.09
60+
934
63.1
0.11
a. Calculate and interpret a confidence interval at confidence level approximately 95% for the
difference between population mean height for the younger women and that for the older
women.
b. Let 𝜇1 denote the population mean height for those aged 20-39 and 𝜇2 denote the
population mean height for those ages 60 and older. Interpret the hypotheses 𝐻0 : 𝜇1 −
𝜇2 = 1, 𝐻𝑎 : 𝜇1 − 𝜇2 > 1, and then carry out a test of these hypotheses at significance level
0.001 using the rejection region approach.
c. What is the P-value for the test you carried out in (b)? Based on this P-value, would you
reject the null hypothesis at any reasonable level of significance?
d. What hypotheses would be appropriate if 𝜇1 referred to the older age group and 𝜇2 referred
to the younger age group and you wanted to see if there was compelling evidence for
concluding that the population mean height for younger women exceeded that for older
women by more than one inch?
2. Tensile strength tests were carried out on two different grades of wire rod, resulting in the
accompanying data.
𝒌𝒈
Grade
Sample Size
Sample Standard
Sample Mean (𝒎𝒎𝟐)
Deviation
AISI 1064
𝑚 = 129
𝑥̅ = 107.6
𝑠1 = 1.3
AISI 1078
𝑛 = 129
𝑦̅ = 123.6
𝑠2 = 2.0
a. Does the data provide compelling evidence for concluding that true average strength for the
𝑘𝑔
1078 grade exceeds that for the 1064 grade by more than 10 𝑚𝑚2? Test the appropriate
hypotheses using the P-value approach.
b. Estimate the difference between true average strengths for the two grades in a way that
provides information about precision and reliability.
3. Determine the number of degrees of freedom for the two-sample t-test or confidence interval in
each of the following situations.
a. 𝑚 = 10, 𝑛 = 10, 𝑠1 = 5.0, 𝑠2 = 6.0
b. 𝑚 = 10, 𝑛 = 15, 𝑠1 = 5.0, 𝑠2 = 6.0
c. 𝑚 = 10, 𝑛 = 15, 𝑠1 = 2.0, 𝑠2 = 6.0
d. 𝑚 = 12, 𝑛 = 24, 𝑠1 = 5.0, 𝑠2 = 6.0
4. Suppose 𝜇1 and 𝜇2 are true mean stopping distances at 50 mph for cars for a certain type
equipped with two different types of braking systems. Use the two-sample t-test at significance
level 0.01 to test 𝐻0 : 𝜇1 − 𝜇2 = −10, 𝐻𝑎 : 𝜇1 − 𝜇2 < −10 for the following data: 𝑚 = 6, 𝑥̅ =
115.7, 𝑠1 = 5.03, 𝑛 = 6, 𝑦̅ = 129.3, 𝑠2 = 5.38
5. Low back pain (LBP) is a serious health problem in many industrial settings. An article from 1995
reported the accompanying summary data on lateral range of motion (degrees) for a sample of
workers without a history of LBP and another sample with a history of this malady.
Condition
Sample Size
Sample Mean
Sample Standard
Deviation
No LBP
28
91.5
5.5
LBP
31
88.3
7.8
Calculate a 90% confidence interval for the difference between population mean extent of
lateral motion for the two conditions. Does the interval suggest that population mean lateral
motion for the two conditions differs? Is the message different is a confidence level of 95% is
used?
6. Scientists and engineers frequently wish to compare two different techniques for measuring or
determining the value of a variable. In such situations, interest centers on testing whether the
mean difference in measurements is zero. An article reports the accompanying data on amount
of milk ingested by each of 14 randomly selected infants.
1
2
3
4
5
6
7
DD
1509
1418
1561
1556
2169
1760
1098
Method
TW
1498
1254
1336
1565
2000
1318
1410
Method
Difference
11
164
225
-9
169
442
-312
8
9
10
11
12
13
14
DD
1198
1479
1281
1414
1954
2174
2058
Method
TW
1129
1342
1124
1468
1604
1722
1518
Method
Difference
69
137
157
-54
350
452
540
a. Is it plausible that the population of differences is normal?
b. Does it appear that the true average difference between intake values measured by the two
methods is something other than zero? Determine the P-value of the test, and use it to
reach a conclusion at significance level 0.05.
7. Is someone who switches brands because of a financial inducement less likely to remain loyal
than someone who switches without inducement? Let 𝑝1 and 𝑝2 denote the true proportion of
switchers to a certain brand with and without inducement, respectively, who subsequently
make a repeat purchase. Test 𝐻0 : 𝑝1 − 𝑝2 = 0, 𝐻𝑎 : 𝑝1 − 𝑝2 < 0 using 𝛼 = 0.01 and the
following data:
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒔𝒖𝒄𝒄𝒆𝒔𝒔𝒆𝒔 = 𝟑𝟎
𝒎 = 𝟐𝟎𝟎
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 = 180
𝒏 = 𝟔𝟎𝟎
8. Researchers sent 5000 resumes in response to job ads that appeared in the Boston Globe and
Chicago Tribune. The resumes were identical except that 2500 of them had “white” sounding
names like Brett and Emily, and 2500 had “black” sounding names like Tamika and Rasheed.
The resumes of the first type elicited 250 responses and the resumes of the second type only
167 responses. Does this data strongly suggest that a resume with a “black” name is less likely
to result in a response than is a resume with a “white” name?
9. In an experiment to compare the tensile strengths of 𝐼 = 5 different types of copper wire 𝐽 = 4
samples of each type were used. The between-samples and within-samples estimates of 𝜎 2
were computed as 𝑀𝑆𝑇𝑟 = 2673.3, 𝑀𝑆𝐸 = 1094.2 respectively.
a. Use the F test at level 0.05 to test 𝐻0 : 𝜇1 = 𝜇2 = 𝜇3 = 𝜇4 = 𝜇5 , 𝐻𝑎 : 𝜇𝑖 ≠ 𝜇𝑗 , 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑖 ≠
𝑗.
b. What can be said about the P-value for the test?
10. It is common practice in many countries to destroy (shred) refrigerators at the end of their
useful lives. In this process material from insulating foam may be released into the atmosphere.
An article gave the following data produced by 4 different manufacturers.
1.
30.4, 29.2
2.
27.7, 27.1
3.
27.1, 24.8
4.
25.5, 28.8
Does it appear that true average foam density is not the same for all these manufacturer? Carry
out an appropriate test of hypotheses by obtaining as much P-value information as possible, and
summarize your analysis in an ANOVA table.
11. An experiment to compare the spreading rates of five different brands of yellow interior latex
paint available in a particular area used 4 gallons (J=4) of each paint. The sample average
spreading rates (ft2/gal) for the five brands were 𝑥̅1 = 462.0, 𝑥̅2 = 512.8, 𝑥̅3 = 437.5, 𝑥̅4 =
469.3, 𝑥̅5 = 532.1. The computed value of F was found to be significant at the level 𝛼 = 0.05.
With 𝑀𝑆𝐸 = 272.8. Use Tukey’s procedure to investigate significant differences in the true
average spreading rates between brands. Use the underscoring method to illustrate your
conclusions and write a paragraph summarizing the results.
a. Repeat this problem by changing 𝑥̅3 = 427.5.
b. Repeat this problem by also changing 𝑥̅2 = 502.8.