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```					                         CHAPTER 7 – Confidence Intervals
7.1   There is an entire population of possible sample means. A confidence interval is constructed
to be confident that the true value of μ is contained in that interval.

7.2   Explanations will vary.

7.3   The higher the confidence level, the wider the confidence interval. The larger the sample, the
shorter the confidence interval.

a. longer

b. shorter

c. shorter

d. longer

7.4   a.   1.96

b.   2.575

c.   3

d.   1.28

e.   2.17

f.   1.75

            s 
7.5    x  z / 2   
             n

            2 
a.   50  (1.96)
       [49.608, 50.392]

            100 

             2 
b.   50  (2.575)
       [49.485, 50.515]

             100 

            2 
c.   50  (2.17)
       [49.566, 50.434]

            100 

            2 
d.   50  (1.28)
       [49.744, 50.256]

            100 

       2 
e.   50  3
       [49.400, 50.600]

       100 

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Chapter 7: Confidence Intervals

              1.6438 
7.6   a.   50.575  1.96
          [50.066, 51.084]

              40 
               1.6438 
50.575  2.575
          [49.906, 51.244]

               40 

b.   Yes, 95% interval is above 50.

c.   No, 99% interval extends below 50.

d.   Fairly confident, only 2.5% chance the mean is less than 50.066.

            2.475 
7.7   a.   5.46  1.96
         [4.975, 5.945]

            100 
             2.475 
5.46  2.575
         [4.823, 6.097 ]

             100 

b.   Yes, 95% interval is below 6.

c.   No, 99% interval extends above 6.

d.   Explanations will vary.

             2.6424 
7.8   a.   42.95  1.96
          [42.308, 43.592]

             65 
              2.6424 
42.95  2.575
          [42.106, 43.794]

              65 

b.   Yes, 95% interval is above 42.

c.   Yes, 99% interval is above 42.

d.   Very confident based on the 99% confidence interval.

            8.70 
7.9   a.   5.68  2.33
        [3.653, 7.707 ]

            100 

b.   3.653

            1.42 
7.10   a.   7.46  1.96
        [7.321, 7.599]

            400 

             1.55 
b.   12.44  1.96
        [12.288,12.592]

             400 

c.   Yes, the minimum value in the confidence interval is greater.

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Chapter 7: Confidence Intervals

            32.83 
7.11   a.    82.6  1.96
         [76.165, 89.035]

            100 

          37.18 
b.    93  1.96
         [85.713,100.287 ]

          100 

c.   Mean audit delay for public owner controlled companies appears to be shorter.

            .71 
7.12   a.    3.3  2.575
       [3.235, 3.365]

            800 

            .66 
b.    4.3  2.575
       [4.224, 4.376]

            500 

c.   Mean number of dealers visited by late replacement buyers appears to be higher.

7.13   a.   Decreases

b.   Decrease

7.14   When the sample is small (n < 30),  is unknown, and the population is normal or
approximately normal.

7.15   t.10  1.363, t.025  2.201, t.001  4.025

7.16   t.05  1.943, t.005  3.707, t.0005  5.959

           5 
7.17   a.               
72  2.228    
  [68.641, 75.359]
           11 

           5 
b.    72  3.169
      [67.223, 76.777 ]

           11 

           5 
c.    72  1.372
      [69.932, 74.068]

           11 

           5 
d.    72  1.812
      [69.268, 74.732]

           11 

           5 
e.    72  2.764
      [67.833, 76.167 ]

           11 

           5 
f.    72  4.144
      [65.753, 78.247 ]

           11 

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Chapter 7: Confidence Intervals

          1.8257 
7.18   a.   6  2.447
          [4.311, 7.689]

             7 
          1.8257 
6  3.707
          [3.442, 8.558]

             7 

b.   Can be 95% confident the claim is true. Cannot be 99% confident the claim is true.

7.19   No, confidence interval below 1 year. Sample size too small.

             1.57 
7.20   a.   13.8  2.093
        [13.065,14.535]

             20 

b.   Yes, 95% interval is below 17.

             .31 
7.21   a.   1.63  2.201
       [1.433,1.827]

             12 

            .23 
b.   .89  2.201
       [.744,1.036]

            12 

c.   Average annual expenses for stock funds appear to be higher.

            19.6469 
7.22   a.   811  2.776
           [786.609, 835.391]

                5 

b.   Yes, the 95% interval is above 750.

               1.6438 
7.23   50.575  2.023
          50.049,51.101 Slightly longer.

               40 

7.24   t-based 95% CI: [4.969, 5.951]
Slightly longer
95% CI for median: [4.574, 5.877]

7.25   See page 273 in text. “Error” refers to the error of our sample mean as an estimate of the
population mean.

7.26   See page 273 in text. B will be the half length of the 99% confidence interval.

7.27   Because  is unknown and must be estimated by s obtained from a preliminary sample.

z 
2

7.28   n  /2 
 B 
2
 2(10) 
a.   n         400 units
 1 

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Chapter 7: Confidence Intervals

                   10 
b.   x  2      295  2
       [294, 296]

        n           400 
error bound = 1
2
 1.96(32.83) 
7.29   a.    n               259 companies
      4      
2
 2.575(32.83) 
b.    n                447 companies
      4       
2
 2.575(.66) 
7.30   a.    n              1806 buyers
    .04     
2
 3(.66) 
b.    n          1569 buyers
 .05 
2
 2.776(19.6469 ) 
7.31   a.    n                   47 trial runs
        8        
2
 4.604 (19.6469 ) 
b.    n                    328 trial runs
         5        
2
 2.201(.31) 
7.32   n              21 stock funds
    .15     
2
 2.093(1.57 ) 
7.33   n                44 stabilization fimes
     .5       

7.34   a.   the proportion of population units that fall in a category of interest

b.                              ˆ
p = population proportion, p = sample proportion

c.   it refers to the error bound in a 95% confidence interval.

7.35   a.   p = .5

b.   p = .3

c.   p = .8

7.36   a.   No; (np) and n(1-p) are not both greater than 5:
np = (.1)(30) = 3
n(1-p) = 30(.9) = 27

b.   Yes; (np) and n(1-p) are both greater than 5:
np = (.1)(100) = 10
n(1-p) = 100(.9) = 90

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Chapter 7: Confidence Intervals

c.   Yes; (np) and n(1-p) are both greater than 5:
np = (.5)(50) = 25
n(1-p) = 50(.5) = 25

d.   Yes; (np) and n(1-p) are both greater than 5:
np = (.8)(400) = 320
n(1-p) = 400(.2) = 80

e.   No; (np) and n(1-p) are not both greater than 5:
np = (.9)(30) = 27
n(1-p) = 30(.1) = 3

f.   No; (np) and n(1-p) are not both greater than 5:
np = (.99)(200) = 198
n(1-p) = 200(.01) = 2

7.37   95% C.I.: z / 2  1.96
98% C.I.: z / 2  2.33
99% C.I.: z / 2  2.575

            (.4)(.6) 
a.   .4  z / 2          

               99  ; [.303, .497], [.285, .515], [.273, .527]


            (.1)(.9) 
b.   .1  z / 2          

              299  ; [.066, .134], [.060, .140], [.055, .145]


            (.9)(.1) 
c.   .9  z / 2          

               99  ; [.841, .959], [.830, .970], [.822, .978]


            (.6)(.4) 
d.   .6  z / 2          

               49  ; [.463, .737], [.437, .763], [.420, .780]


7.38        Social Security :
(.23)(.77 )
.23  1.96               .23  .0257  [.2043,.2557 ]
1030
Education :
(.21)(.79)
.21  1.96               .21  .0249  [.1851,.2349 ]
1030

              (.54146 )(.45854 ) 
7.39   a.   .54146  1.96                      [.473, .610]

                    204          


b.   No, the interval extends below .5.

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Chapter 7: Confidence Intervals

           (.41)(.59) 
7.40   a.   .41  1.96              [.3795, .4405]

              999    

b.   37.95%

(.42)(.58)
7.41   a.   .42  2.575                .42  .04  [.38,.46]
1030
No

(.60)(.40)
b.   .60  1.96                .60  .03  [.57,.63]
1030
Yes

c.   95% error bound is .03

142
7.42   a.    p
ˆ         .355
400
              (.355)(.645) 
.355  1.96                  [.308, .402]

                  399      


122
b.    p
ˆ         .244
500
              (.244)(.756) 
.244  1.96                  [.206, .282]

                  499      


c.   Yes, the U.K. 95% confidence interval is above the maximum value in the confidence
interval for the U.S.

            (.67)(.33) 
7.43   a.   .67  2.575              [.611, .729]

               417    

b.   Yes, the interval is above .6.

(.20)(.80)
7.44   .20  1.96                .20  .0166  [.1834,.2166 ]
2220
Yes, entire interval is below .25

515
7.45   p
ˆ         .831
620

           (.831)(.169) 
a.   .831  2.0                [.801, .861]

               619      


b.   Yes, the interval is above .75.

316
7.46   p
ˆ         .79
400

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Chapter 7: Confidence Intervals

            (.79)(.21) 
a.   .79  2.575              [.737, .843]

               399    

b.   Yes, the interval is below .95.

10
7.47   a.    p
ˆ         .02
500
             (.02)(.98) 
.02  1.96                [.0077 , .0323]

                499    

27
b.    p
ˆ         .054
500
              (.054)(.946) 
.054  1.96                  [.034, .074]

                  499      


c.   Yes, the confidence interval is higher.
2
 1.96 
7.48   n  (.57 )(.43)        2353 .9  2354 adults
 .02 
2                         2
 2.575               2.575 
7.49   n .737 (1 – .737 )         737 (.263)         1428 .0229 1429 televisions
 .03                 .03 
2
 1.96 
7.50   n  .5(1 – .5)        .25(38,416 )  9604 voters
 .01 

7.51   Sum of the values of all of the population measurements.

7.52   Finite population correction is very close to one.

            s               168 
7.53   a.    x  z / 2     532  1.96        [532  17.601]  [\$514 .399, \$549 .601]
             n               350 

b.   95% C.I. for = [\$5,375,983.95, \$5,743,880.05]
Point estimate = \$5,559,932

c.   Claim is very doubtful.

31
7.54   a.    p
ˆ           .31
100
               p(1 – p)  N – n   
ˆ     ˆ                            (.31)(.69)  1323 – 100  
 p  z / 2
ˆ                                .31  1.96                           [.222, .398]

                 n –1  N                        99      1323       

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Chapter 7: Confidence Intervals

b.     Np  1323(.31)  410
ˆ
               p(1 – p)  N – n  
ˆ     ˆ
 Np  z / 2 N
ˆ                              [294, 526]

                 n –1  N       

c.    \$2940, yes

7.55   a.     Nx  687(4.3)  2954
               s    N –n                     1.26  687 – 50 
 Nx  z / 2 N            2954  1.96(687 )
      
            [2723, 3185]

                n    N  
                     50      687   

b.    Yes, the interval is above 2500; no, the interval extends below 3000.

10
7.56   p
ˆ         .3125, Np  600(.3125)  187.5
ˆ
32

                      .3125 (.6875 )  600  32  
187.5  1.96(600)                                 187.5  95.25  [92.25, 282.75] or [92, 283]

                          32  1      600      
2                   2
 B               100       
7.57            Z N    (1.96)(687 )   .0055
a. D                             
  /2                      
N   2
(687 )(1.5876 )
n                                               203 .46  204
( N  1) D   2
(686 )(.0055 )  1.5876
2
 B     
2
 
.04 
b. D  
Z                  .000416
  /2       1.96 
1323(.31)(.69)
n                                 370 .16  371
(1322 )(.000416 )  (.31)(.69)

7.58   Use stratified random sampling when population consists of two or more subpopulations that
differ in terms of the study. The subpopulations are called strata and should be defined
so that the between-strata variability is larger than the variability within each stratum for
the characteristic under study.

7.59   Use cluster sampling when samples are taken from large geographical regions. The
population is “clustered” into subpopulations.

7.60   a. Some of the population is not represented.

b. Some of the chosen sample does not respond.

c. Responses may not be truthful.

7.61   1853/100=18.53 Randomly select a company from the first 18 companies of the list. Use that
company and every 18th company until 100 have been chosen.

7.62   Explanations will vary.

97
Chapter 7: Confidence Intervals

7.63   Explanations will vary.

7.64   A tolerance interval is supposed to contain a specified percentage of individual population
measurements.

7.65   The population mean, μ.

7.66   The tolerance interval contains individual measurements which includes the high and low
measurements that are averaged out in the confidence interval of the mean.

7.67   68.26% tolerance interval:     50.575  1.6438 = [48.9312, 52.2188]

95.44% tolerance interval:     50.575  3.2876 = [47.2874, 53.8626]

99.73% tolerance interval:     50.575  4.9314 = [45.6436, 55.5064]

95% confidence interval: 50.575  1.96(1.6438        40 )  [50.0656 , 51.0844 ]

7.68   68.26% tolerance interval:     5.46  2.475 = [2.985, 7.935]

95.44% tolerance interval:     5.46  4.95 = [0.51, 10.41]

99.73% tolerance interval:     5.46  7.425 = [-1.965, 12.885]

95% confidence interval:     5.46  1.96(2.475     100 )  [4.9749, 5.9451]

7.69   68.26% tolerance interval:     42.95  2.6424 = [40.3076, 45.5924]

95.44% tolerance interval:     42.95  5.2848 = [37.6652, 48.2348]

99.73% tolerance interval:     42.95  7.9272 = [35.0228, 50.8772]

95% confidence interval:     42.95  1.96(2.6424      65 )  [42.3076, 43.5924 ]

             35.72 
7.70   a.   68.04  1.96
         [63.612, 72.468]

             250 

             34.87 
b.   56.74  1.96
         [52.310, 61.170]

             238 

c.   Yes, interval is below the lowest value in the confidence interval.

142
7.71   a.   p
ˆ          .3054
465
               (.3054 )(.6946 ) 
.3054  1.96                      [.2635, .3473]

                    464         


b.   Yes; yes

98
Chapter 7: Confidence Intervals

2
n  p(1 p)  /2 
z
c.
 B 
2
 1.96 
n  (.3473)(1 – .3473)       = 967.58 968
 .03 

(.61)(.39)
7.72   a.   .61  1.96               .61  .038  [.572,.648]
621
Yes

b.   95% error bound is .038

 8 
7.73   Np  57,532
ˆ                    2301 .28
 200 
                 p(1 – p)  N – n   
ˆ      ˆ                                         (.04)(.96)  57,532 – 200  
 Np  z / 2 N
ˆ                                2301 .28  1.96(57,532 )                          

                   n –1  N                                     199  57,532  
              
= [737.60, 3864.96]

 6.02 
7.74   a.   57.8  1.96
        [56.489, 59.111]

 81 

Yes, interval is below 60.
2                  2
 Z s  1.96(6.02) 
b.   n /2              139 .22  140
 B        1      

7.75   a.   Nx  257 (75,162.70)  \$19,316,814
               s       N –n                            28,865 .04  257 – 40 
 Nx  z / 2 N               19,316,814  2.575(257 )
            
          

                n       N  
                                40        257   
 [\$16,541,476, \$22,092,152 ]

b.   \$22,092,152

150
7.76   a.   p
ˆ           .746
201
              p(1 – p)  
ˆ     ˆ                   (.746)(.254) 
 p  z / 2
ˆ                       .746  1.96                [.686, .806]

                n 1  
                   200      

No, the proportion could be below .7.

          s                 6.11 
b.    x  1.96
      4.88  1.96
                        [3.902, 5.858]

          n                 150 
Yes, the interval is above 3.

             3.7432 
7.77   a.   26.22  1.96
          [25.182, 27.258]

             50 

99
Chapter 7: Confidence Intervals

b.   Yes, not much more than 25

              1.64 
7.78   a.   2.73  2.977        [1.469%, 3.991%]

              15 
Yes, the interval is below 5%.

                25.37 
b.   34.76  2.977          [15.259%, 54.261%]

                15 
This interval is wide because s is large and n is small; increase the sample size.

          s 
7.79   a.   95% C.I.   x  1.96   
           n
Fixed annuities: [7.689%, 7.971%]
Domestic large-cap stocks: [9.215%, 17.625%]
Domestic mid-cap stocks: [9.919%, 20.141%]
Domestic small-cap stocks: [16.481%, 28.539%]

b.   Fixed annuities: differ from 8.31%
Domestic large-cap stocks: does not differ from 11.71%
Domestic mid-cap stocks: does not differ from 13.64%
Domestic small-cap stocks: differs from 14.93%

 41 
7.80   538  1.96
      [524 .42, 551 .58]

 35 

(.64)(.36)
7.81   .64  1.96               .64  .03  [.61,.67]
999

               1.4 
7.82   a.   121.92  2.262
       [\$120.92, \$122.92]

               10 

               1.99 
114.73  2.262
        [\$113.31, \$116.15]

               10 

b.   Yes, the mean cost at Miller’s seems higher.

100
Chapter 7: Confidence Intervals

Internet Exercise
7.83

Descriptive Statistics
Variable: price

Anderson-Darling Normality Test
A-Squared:              4.944
P-Value:                0.000

Mean                  1062.74
StDev                  380.44
Variance               144732
Skewness              1.37540
Kurtosis              1.44746
N                         117
600   900      1200      1500      1800     2100
Minimum                540.00
1st Quartile           776.50
Median                 960.00
3rd Quartile          1217.50
99% Confidence Interval for Mu                  Maximum               2150.00
99% Confidence Interval for Mu
970.63               1154.84
850           950                1050             1150     99% Confidence Interval for Sigma
324.97                456.62
99% Confidence Interval for Median
99% Confidence Interval for Median
877.11               1049.91

MINITAB Graphical Summary

101

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