A random sample of 28 items is drawn from a population whose standard deviation is unknown. The sample mean is x¯x¯ = 790 and the sample standard deviation is s = 15. Use Appendix D to find the values of Student’s t.(a) Construct an interval estimate of μ with 99% confidence.(b) Construct an interval estimate of μ with 99% confidence, assuming that s = 30.(c) Construct an interval estimate of μ with 99% confidence, assuming that s = 60. (Round your answers to 3 decimal places.) APPENDIXSTUDENT'St CRITICAL VALUESThis table shows the t-value that defines the area for the stated degrees of freedom (d.f.).Confidence LevelConfidence Level.80.90.95.98.99.80.90.95.98.99Significance Level for Two-Tailed TestSignificance Level for Two-Tailed Test.20.10.05.02.01.20.10.05.02.01Significance Level for One-Tailed TestSignificance Level for One-Tailed Testd.f..10.05.025.01.005d.f..10.05.025.01.00513.0786.31412.70631.82163.65736 1.3061.6882.0282.4342.7191.8862.9204.3036.9659.925371.305 1.687 2.0262.4312.7153.1.6382.3533.1824.5415.841381.3041.686 2.0242.4292.71241.5332.1322.7763.7474.604391.3041.6852.0232.4262.7081.4762.0152.5713.3654.032401.3031.6842.0212.4232.70461.4401.9432.4473.1433.707411.3031.6832.0202.4212.70171.4151.8952.3652.9983.49942 1.3021.6822.0182.4182.69881.3971.8602.3062.8963.355431.3021.6812.0172.4162.69591.3831.8332.2622.8213.250441.3011.6802.0152.4142.692101.3721.8122.2282.7643.169451.3011.6792.0142.4122.690111.3631.7962.2012.7183.106461.3001.6792.0132.4102.687121.3561.7822.1792.6813.055471.3001.6782.0122.4082.685131.3501.7712.1602.6503.012481.2991.6772.0112.4072.682141.3451.7612.1452.6242.97749 1.2991.6772.0102.4052.680151.3411.7532.1312.6022.947501.2991.6762.0092.4032.678161.3371.7462.1202.5832.92155 1.297 1.6732.0042.3962.668171.3331.7402.1102.5672.89860 1.2961.6712.0002.390 2.660181.3301.7342.1012.5522.878651.2951.6691.9972.3852.654191.3281.7292.0932.5392.86170 1.2941.6671.9942.3812.648201.3251.7252.0862.5282.84575 1.2931.6651.9922.3772.64321 1.3231.7212.0802.5182.83180 1.2921.6641.9902.3742.639221.3211.7172.0742.5082.819851.2921.6631.9882.3712.635231.3191.7142.0692.5002.807901.291 1.6621.9872.3682.632241.3181.7112.0642.4922.797951.2911.6611.9852.3662.629251.3161.7082.0602.4852.7871001.2901.6601.9842.3642.626261.3151.7062.0562.4792.7791101.289 1.6591.9822.3612.621271.3141.7032.0522.4732.7711201.2891.6581.9802.3582.617281.3131.7012.0482.4672.7631301.2881.6571.9782.3552.614291.3111.6992.0452.4622.7561401.288 1.6561.9772.3532.611301.3101.6972.0422.4572.7501501.2871.6551.9762.3512.609311.3091.6962.0402.4532.7441.2821.6451.9602.3262.57600321.3091.6942.0372.4492.738331.3081.6922.0352.4452.733341.3071.6912.0322.4412.728351.3061.6902.0302.4382.724Note: As n increases, critical values of Student's t approach the z-values in the last line of this table. A common rule of thumb is to use z when n> 30, but that is not conservative.

Given: n= 28.0 , x̄= 790.0 s= 15.0 degree of freedom=n-1= 27.0 At 99.0 % confidence level with d.o.f.= 27.0 ,t-score= 2.771a)So, 99.0 % confidence interval for population mean is given by- µ=x̄ ± t*(s/√(n)) µ= 790.0 ± 2.771 *( 15.0 / √( 28.0 )) µ= 790.0 ± 7.855 lower limit= 790.0 - 7.855 = 782.145 upper limit= 790.0 + 7.855 = 797.855 Hence, 782.145 <µ< 797.855 confidence interval of population mean µ is 782.145 to 797.855 b)s= 30.0 So, 99.0 % confidence interval for population mean is given by- µ=x̄ ± t*(s/√(n)) µ= 790.0 ± 2.771 *( 30.0 / √( 28.0 )) µ= 790.0 ± 15.7101 lower limit= 790.0 - 15.7101 = 774.2899 upper limit= 790.0 + 15.7101 = 805.7101 Hence, 774.290 <µ< 805.710 confidence interval of population mean µ is 774.290 to 805.710c)s= 60.0 So, 99.0 % confidence interval for population mean is given by- µ=x̄ ± t*(s/√(n)) µ= 790.0 ± 2.771 *( 60.0 / √( 28.0 )) µ= 790.0 ± 31.4202 lower limit= 790.0 - 31.4202 = 758.5798 upper limit= 790.0 + 31.4202 = 821.4202 Hence, 758.580 <µ< 821.420 confidence interval of population mean µ is 758.580 to 821.420

Answered: A random sample of 28 items is drawn…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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Question

A random sample of 28 items is drawn from a population whose standard deviation is unknown. The sample mean is x¯x¯ = 790 and the sample standard deviation is s = 15. Use Appendix D to find the values of Student’s t.

(a) Construct an interval estimate of μ with 99% confidence.

(b) Construct an interval estimate of μ with 99% confidence, assuming that s = 30.

(c) Construct an interval estimate of μ with 99% confidence, assuming that s = 60. (Round your answers to 3 decimal places.)

APPENDIX
STUDENT'St CRITICAL VALUES
This table shows the t-value that defines the area for the stated degrees of freedom (d.f.).
Confidence Level
Confidence Level
.80
.90
.95
.98
.99
.80
.90
.95
.98
.99
Significance Level for Two-Tailed Test
Significance Level for Two-Tailed Test
.20
.10
.05
.02
.01
.20
.10
.05
.02
.01
Significance Level for One-Tailed Test
Significance Level for One-Tailed Test
d.f.
.10
.05
.025
.01
.005
d.f.
.10
.05
.025
.01
.005
1
3.078
6.314
12.706
31.821
63.657
36 1.306
1.688
2.028
2.434
2.719
1.886
2.920
4.303
6.965
9.925
37
1.305 1.687 2.026
2.431
2.715
3.
1.638
2.353
3.182
4.541
5.841
38
1.304
1.686 2.024
2.429
2.712
4
1.533
2.132
2.776
3.747
4.604
39
1.304
1.685
2.023
2.426
2.708
1.476
2.015
2.571
3.365
4.032
40
1.303
1.684
2.021
2.423
2.704
6
1.440
1.943
2.447
3.143
3.707
41
1.303
1.683
2.020
2.421
2.701
7
1.415
1.895
2.365
2.998
3.499
42 1.302
1.682
2.018
2.418
2.698
8
1.397
1.860
2.306
2.896
3.355
43
1.302
1.681
2.017
2.416
2.695
9
1.383
1.833
2.262
2.821
3.250
44
1.301
1.680
2.015
2.414
2.692
10
1.372
1.812
2.228
2.764
3.169
45
1.301
1.679
2.014
2.412
2.690
11
1.363
1.796
2.201
2.718
3.106
46
1.300
1.679
2.013
2.410
2.687
12
1.356
1.782
2.179
2.681
3.055
47
1.300
1.678
2.012
2.408
2.685
13
1.350
1.771
2.160
2.650
3.012
48
1.299
1.677
2.011
2.407
2.682
14
1.345
1.761
2.145
2.624
2.977
49 1.299
1.677
2.010
2.405
2.680
15
1.341
1.753
2.131
2.602
2.947
50
1.299
1.676
2.009
2.403
2.678
16
1.337
1.746
2.120
2.583
2.921
55 1.297 1.673
2.004
2.396
2.668
17
1.333
1.740
2.110
2.567
2.898
60 1.296
1.671
2.000
2.390 2.660
18
1.330
1.734
2.101
2.552
2.878
65
1.295
1.669
1.997
2.385
2.654
19
1.328
1.729
2.093
2.539
2.861
70 1.294
1.667
1.994
2.381
2.648
20
1.325
1.725
2.086
2.528
2.845
75 1.293
1.665
1.992
2.377
2.643
21 1.323
1.721
2.080
2.518
2.831
80 1.292
1.664
1.990
2.374
2.639
22
1.321
1.717
2.074
2.508
2.819
85
1.292
1.663
1.988
2.371
2.635
23
1.319
1.714
2.069
2.500
2.807
90
1.291 1.662
1.987
2.368
2.632
24
1.318
1.711
2.064
2.492
2.797
95
1.291
1.661
1.985
2.366
2.629
25
1.316
1.708
2.060
2.485
2.787
100
1.290
1.660
1.984
2.364
2.626
26
1.315
1.706
2.056
2.479
2.779
110
1.289 1.659
1.982
2.361
2.621
27
1.314
1.703
2.052
2.473
2.771
120
1.289
1.658
1.980
2.358
2.617
28
1.313
1.701
2.048
2.467
2.763
130
1.288
1.657
1.978
2.355
2.614
29
1.311
1.699
2.045
2.462
2.756
140
1.288 1.656
1.977
2.353
2.611
30
1.310
1.697
2.042
2.457
2.750
150
1.287
1.655
1.976
2.351
2.609
31
1.309
1.696
2.040
2.453
2.744
1.282
1.645
1.960
2.326
2.576
00
32
1.309
1.694
2.037
2.449
2.738
33
1.308
1.692
2.035
2.445
2.733
34
1.307
1.691
2.032
2.441
2.728
35
1.306
1.690
2.030
2.438
2.724
Note: As n increases, critical values of Student's t approach the z-values in the last line of this table. A common rule of thumb is to use z when n
> 30, but that is not conservative.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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