A research team is working on a project to study the time (in seconds) for high school male runners to finish a 400-meter race. Jimmy, a junior researcher in the team, has randomly selected a sample of 25 male runners from a high school and the time (in seconds) for each of them to complete a 400-meter race was recorded. The sample mean running time was 53 seconds. It is assumed that the running time in a 400-meter race follows a normal distribution with a population standard deviation of 5.5 seconds. (a) Give a point estimate of the population mean running time for a 400-meter race. (b) Calculate the sampling error at 95% confidence level. (c) Construct a 95% confidence interval estimate of the population mean running time for a 400-meter race.
A research team is working on a project to study the time (in seconds) for high school male runners to finish a 400-meter race. Jimmy, a junior researcher in the team, has randomly selected a sample of 25 male runners from a high school and the time (in seconds) for each of them to complete a 400-meter race was recorded. The sample mean running time was 53 seconds. It is assumed that the running time in a 400-meter race follows a normal distribution with a population standard deviation of 5.5 seconds. (a) Give a point estimate of the population mean running time for a 400-meter race. (b) Calculate the sampling error at 95% confidence level. (c) Construct a 95% confidence interval estimate of the population mean running time for a 400-meter race.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
A research team is working on a project to study the time (in seconds) for high school male runners to finish
a 400-meter race. Jimmy, a junior researcher in the team, has randomly selected a sample of 25 male runners
from a high school and the time (in seconds) for each of them to complete a 400-meter race was recorded.
The sample mean running time was 53 seconds. It is assumed that the running time in a 400-meter race
follows a
(a) Give a point estimate of the population mean running time for a 400-meter race.
(b) Calculate the sampling error at 95% confidence level.
(c) Construct a 95% confidence
(d) If instead of 25, 80 male runners are selected for the study, what is the sampling error at 95% confidence
level? When the 95% confidence interval is constructed based on a sample with 80 male runners, would
you expect the interval will be (I) narrower, (II) wider, or (III) the same width as the confidence interval
constructed in part (c)?
(e) After discussion with the research director, Jimmy is asked to work on the research again. This time,
he needs to ensure the difference between the point estimate and the true population mean be within
± 0.8 seconds at 99% confidence level. To fulfil the requirement, how large should be the minimum
sample size?
Transcribed Image Text:The entries in Table I are the probabilities that a random variable having the
standard normal distribution will take on a value between 0 andz. They are given
by the area of the gray region under the curve in the figure.
TABLE I
NORMAL-CURVE AREAS
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.0000
0.0040
0.0080
0.0120
0.0160
0.0199
0.0239
0.0279
0.0319
0.0359
0.1
0.0398
0.0438
0.0478
0.0517
0.0557
0.0596
0.0636
0.0675
0.0714
0.0753
0.2
0.0793
0.0832
0.0871
0.0910
0.0948
0.0987
0.1026
0.1064
0.1103
0.1141
0.3
0.1179
0.1217
0.1255
0.1293
0.1331
0.1368
0.1406
0.1443
0.1480
0.1517
0.4
0.1554
0.1591
0.1628
0.1664
0.1700
0.1736
0.1772
0.1808
0.1844
0.1879
0.5
0.1915
0.1950
0.1985
0.2019
0.2054
0.2088
0.2123
0.2157
0.2190
0.2224
0.6
0.2257
0.2291
0.2324
0.2357
0.2389
0.2422
0.2454
0.2486
0.2517
0.2549
0.7
0.2580
0.2611
0.2642
0.2673
0.2704
0.2734
0.2764
0.2794
0.2823
0.2852
0.8
0.2881
0.2910
0.2939
0.2967
0.2995
0.3023
0.3051
0.3078
0.3106
0.3133
0.9
0.3159
0.3186
0.3212
0.3238
0.3264
0.3289
0.3315
0.3340
0.3365
0.3389
1.0
0.3413
0.3438
0.3461
0.3485
0.3508
0.3531
0.3554
0.3577
0.3599
0.3621
1.1
0.3643
0.3665
0.3686
0.3708
0.3729
0.3749
0.3770
0.3790
0.3810
0.3830
0.3907
0.4082
1.2
0.3849
0.3869
0.3888
0.3925
0.3944
0.3962
0.3980
0.3997
0.4015
1.3
0.4032
0.4049
0.4066
0.4099
0.4115
0.4131
0.4147
0.4162
0.4177
1.4
0.4192
0.4207
0.4222
0.4236
0.4251
0.4265
0.4279
0.4292
0.4306
0.4319
1.5
0.4332
0.4345
0.4357
0.4370
0.4382
0.4394
0.4406
0.4418
0.4429
0.4441
1.6
0.4452
0.4463
0.4474
0.4484
0.4495
0.4505
0.4515
0.4525
0.4535
0.4545
0.4573
0.4656
0.4582
0.4664
0.4591
0.4671
0.4599
0.4678
0.4608
0.4685
0.4625
0.4699
0.4633
0.4706
1.7
0.4554
0.4564
0.4616
1.8
0.4641
0.4648
0.4692
1.9
0.4713
0.4719
0.4725
0.4732
0.4738
0.4744
0.4750
0.4756
0.4761
0.4767
2.0
0.4772
0.4778
0.4783
0.4788
0.4793
0.4798
0.4803
0.4808
0.4812
0.4817
2.1
0.4821
0.4826
0.4830
0.4834
0.4838
0.4842
0.4846
0.4850
0.4854
0.4857
2.2
0.4861
0.4864
0.4868
0.4871
0.4875
0.4878
0.4881
0.4884
0.4887
0.4890
2.3
0.4893
0.4896
0.4898
0.4901
0.4904
0.4906
0.4909
0.4911
0.4913
0.4916
2.4
0.4918
0.4920
0.4922
0.4925
0.4927
0.4929
0.4931
0.4932
0.4934
0.4936
2.5
0.4938
0.4940
0.4941
0.4943
0.4945
0.4946
0.4948
0.4949
0.4951
0.4952
2.6
0.4953
0.4955
0.4956
0.4957
0.4959
0.4960
0.4961
0.4962
0.4963
0.4964
0.4965
0.4974
0.4969
0.4977
0.4971
0.4979
0.4973
0.4980
2.7
0.4966
0.4967
0.4968
0.4970
0.4972
0.4974
2.8
0.4975
0.4976
0.4977
0.4978
0.4979
0.4981
0.4984
0.4988
2.9
0.4981
0.4982
0.4982
0.4983
0.4984
0.4985
0.4985
0.4986
0.4986
3.0
0.4987
0.4987
0.4987
0.4988
0.4989
0.4989
0.4989
0.4990
0.4990
Also, for z = 4.0, 5.0 and 6.0, the areas are 0.49997, 0.4999997, and 0.499999999.
Transcribed Image Text:The entries in Table II are values for which the area to their right under the
distribution with given degrees of freedom (the gray area in the figure) is equal
to a.
TABLE II
VALUE OFt
d.f.
'0.0s0
l0.025
f0.010
10.005
d.f.
1
6.314
12.706
31.821
63.657
1
2.920
4.303
6.965
9.925
3
2.353
3.182
4.541
5.841
3
4
2.132
2.776
3.747
4.604
4
5
2.015
2.571
3.365
4.032
5
6.
1.943
2.447
3.143
3.707
6.
1.895
2.365
2.998
3.499
7
8
1.860
2.306
2.896
3.355
8.
9
1.833
2.262
2,821
3.250
9
10
1.812
2.228
2.764
3.169
10
11
1.796
2.201
2.718
3.106
11
12
1.782
2.179
2.681
3.055
12
13
1.771
2.160
2.650
3.012
13
14
1.761
2.145
2.624
2.977
14
15
1.753
2.131
2.602
2.947
15
16
1.746
2.120
2.583
2.921
16
17
1.740
2.110
2.567
2.898
17
18
1.734
2.101
2.552
2.878
18
19
1.729
2.093
2.539
2.861
19
20
1.725
2.086
2.528
2.845
20
21
1.721
2,080
2.518
2.831
21
22
1.717
2.074
2.508
2.819
22
23
1.714
2.069
2.500
2.807
23
24
1.711
2.064
2.492
2.797
24
25
1.708
2.060
2.485
2.787
25
26
1.706
2.056
2.479
2.779
26
27
1.703
2.052
2.473
2.771
27
28
1.701
2.048
2.467
2.763
28
29
1.699
2.045
2.462
2.756
29
Inf.
1.645
1.960
2.326
2.576
Inf.
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