Question 1206830: A survey of 45 randomly selected iPhone owners showed that the purchase price has a mean of $426 with a sample standard deviation of $190.
b. Compute the 95% confidence interval for the mean. (Round the final answers to 2 decimal places.)
The confidence interval is between $
and $
.
Appendix B.1
Areas under the Normal Curve
z 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 0.383
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 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
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 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
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
Example:
If z = 1.96, then
P (0 to z) = 0.4750
z 0 1.96
0.4750
AP-56 Chapter 2
AP-56
Appendix B.2
Student’s t Distribution
Confidence Intervals, c
df
80% 90% 95% 98% 99% 99.9%
Level of Significance for One-Tailed Test, α
0.10 0.05 0.025 0.01 0.005 0.0005
Level of Significance for Two-Tailed Test, α
0.20 0.10 0.05 0.02 0.01 0.001
1 3.078 6.314 12.706 31.821 63.657 636.619
2 1.886 2.920 4.303 6.965 9.925 31.599
3 1.638 2.353 3.182 4.541 5.841 12.924
4 1.533 2.132 2.776 3.747 4.604 8.610
5 1.476 2.015 2.571 3.365 4.032 6.869
6 1.440 1.943 2.447 3.143 3.707 5.959
7 1.415 1.895 2.365 2.998 3.499 5.408
8 1.397 1.860 2.306 2.896 3.355 5.041
9 1.383 1.833 2.262 2.821 3.250 4.781
10 1.372 1.812 2.228 2.764 3.169 4.587
11 1.363 1.796 2.201 2.718 3.106 4.437
12 1.356 1.782 2.179 2.681 3.055 4.318
13 1.350 1.771 2.160 2.650 3.012 4.221
14 1.345 1.761 2.145 2.624 2.977 4.140
15 1.341 1.753 2.131 2.602 2.947 4.073
16 1.337 1.746 2.120 2.583 2.921 4.015
17 1.333 1.740 2.110 2.567 2.898 3.965
18 1.330 1.734 2.101 2.552 2.878 3.922
19 1.328 1.729 2.093 2.539 2.861 3.883
20 1.325 1.725 2.086 2.528 2.845 3.850
21 1.323 1.721 2.080 2.518 2.831 3.819
22 1.321 1.717 2.074 2.508 2.819 3.792
23 1.319 1.714 2.069 2.500 2.807 3.768
24 1.318 1.711 2.064 2.492 2.797 3.745
25 1.316 1.708 2.060 2.485 2.787 3.725
26 1.315 1.706 2.056 2.479 2.779 3.707
27 1.314 1.703 2.052 2.473 2.771 3.690
28 1.313 1.701 2.048 2.467 2.763 3.674
29 1.311 1.699 2.045 2.462 2.756 3.659
30 1.310 1.697 2.042 2.457 2.750 3.646
31 1.309 1.696 2.040 2.453 2.744 3.633
32 1.309 1.694 2.037 2.449 2.738 3.622
33 1.308 1.692 2.035 2.445 2.733 3.611
34 1.307 1.691 2.032 2.441 2.728 3.601
35 1.306 1.690 2.030 2.438 2.724 3.591
0 0
1
2 α α α α1
2
Confidence interval - t t 0
Left-tailed test - t t
Right-tailed test
t
Two-tailed test - t t
Confidence Intervals, c
df
80% 90% 95% 98% 99% 99.9%
Level of Significance for One-Tailed Test, α
0.10 0.05 0.025 0.01 0.005 0.0005
Level of Significance for Two-Tailed Test, α
0.20 0.10 0.05 0.02 0.01 0.001
36 1.306 1.688 2.028 2.434 2.719 3.582
37 1.305 1.687 2.026 2.431 2.715 3.574
38 1.304 1.686 2.024 2.429 2.712 3.566
39 1.304 1.685 2.023 2.426 2.708 3.558
40 1.303 1.684 2.021 2.423 2.704 3.551
41 1.303 1.683 2.020 2.421 2.701 3.544
42 1.302 1.682 2.018 2.418 2.698 3.538
43 1.302 1.681 2.017 2.416 2.695 3.532
44 1.301 1.680 2.015 2.414 2.692 3.526
45 1.301 1.679 2.014 2.412 2.690 3.520
46 1.300 1.679 2.013 2.410 2.687 3.515
47 1.300 1.678 2.012 2.408 2.685 3.510
48 1.299 1.677 2.011 2.407 2.682 3.505
49 1.299 1.677 2.010 2.405 2.680 3.500
50 1.299 1.676 2.009 2.403 2.678 3.496
51 1.298 1.675 2.008 2.402 2.676 3.492
52 1.298 1.675 2.007 2.400 2.674 3.488
53 1.298 1.674 2.006 2.399 2.672 3.484
54 1.297 1.674 2.005 2.397 2.670 3.480
55 1.297 1.673 2.004 2.396 2.668 3.476
56 1.297 1.673 2.003 2.395 2.667 3.473
57 1.297 1.672 2.002 2.394 2.665 3.470
58 1.296 1.672 2.002 2.392 2.663 3.466
59 1.296 1.671 2.001 2.391 2.662 3.463
60 1.296 1.671 2.000 2.390 2.660 3.460
61 1.296 1.670 2.000 2.389 2.659 3.457
62 1.295 1.670 1.999 2.388 2.657 3.454
63 1.295 1.669 1.998 2.387 2.656 3.452
64 1.295 1.669 1.998 2.386 2.655 3.449
65 1.295 1.669 1.997 2.385 2.654 3.447
66 1.295 1.668 1.997 2.384 2.652 3.444
67 1.294 1.668 1.996 2.383 2.651 3.442
68 1.294 1.668 1.995 2.382 2.650 3.439
69 1.294 1.667 1.995 2.382 2.649 3.437
70 1.294 1.667 1.994 2.381 2.648 3.435
AP-57
Appendix B.2
Confidence Intervals, c
df
80% 90% 95% 98% 99% 99.9%
Level of Significance for One-Tailed Test, α
0.10 0.05 0.025 0.01 0.005 0.0005
Level of Significance for Two-Tailed Test, α
0.20 0.10 0.05 0.02 0.01 0.001
71 1.294 1.667 1.994 2.380 2.647 3.433
72 1.293 1.666 1.993 2.379 2.646 3.431
73 1.293 1.666 1.993 2.379 2.645 3.429
74 1.293 1.666 1.993 2.378 2.644 3.427
75 1.293 1.665 1.992 2.377 2.643 3.425
76 1.293 1.665 1.992 2.376 2.642 3.423
77 1.293 1.665 1.991 2.376 2.641 3.421
78 1.292 1.665 1.991 2.375 2.640 3.420
79 1.292 1.664 1.990 2.374 2.640 3.418
80 1.292 1.664 1.990 2.374 2.639 3.416
81 1.292 1.664 1.990 2.373 2.638 3.415
82 1.292 1.664 1.989 2.373 2.637 3.413
83 1.292 1.663 1.989 2.372 2.636 3.412
84 1.292 1.663 1.989 2.372 2.636 3.410
85 1.292 1.663 1.988 2.371 2.635 3.409
86 1.291 1.663 1.988 2.370 2.634 3.407
87 1.291 1.663 1.988 2.370 2.634 3.406
88 1.291 1.662 1.987 2.369 2.633 3.405
Confidence Intervals, c
df
80% 90% 95% 98% 99% 99.9%
Level of Significance for One-Tailed Test, α
0.10 0.05 0.025 0.01 0.005 0.0005
Level of Significance for Two-Tailed Test, α
0.20 0.10 0.05 0.02 0.01 0.001
89 1.291 1.662 1.987 2.369 2.632 3.403
90 1.291 1.662 1.987 2.368 2.632 3.402
91 1.291 1.662 1.986 2.368 2.631 3.401
92 1.291 1.662 1.986 2.368 2.630 3.399
93 1.291 1.661 1.986 2.367 2.630 3.398
94 1.291 1.661 1.986 2.367 2.629 3.397
95 1.291 1.661 1.985 2.366 2.629 3.396
96 1.290 1.661 1.985 2.366 2.628 3.395
97 1.290 1.661 1.985 2.365 2.627 3.394
98 1.290 1.661 1.984 2.365 2.627 3.393
99 1.290 1.660 1.984 2.365 2.626 3.392
100 1.290 1.660 1.984 2.364 2.626 3.390
120 1.289 1.658 1.980 2.358 2.617 3.373
140 1.288 1.656 1.977 2.353 2.611 3.361
160 1.287 1.654 1.975 2.350 2.607 3.352
180 1.286 1.653 1.973 2.347 2.603 3.345
200 1.286 1.653 1.972 2.345 2.601 3.340
∞ 1.282 1.645 1.960 2.326 2.576 3.291
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! A survey of 45 randomly selected iPhone owners showed that the purchase price has a mean of $426 with a sample standard deviation of $190.
Compute the 95% confidence interval for the mean. (Round the final answers to 2 decimal places.)
the sample size is 45.
the sample mean is 426.
the sample standard deviation is 190.
the standard error is equal to standard deviation / sqrt(sample size) = 190 / sqrt(45) = 28.3235.
t-score is indicated because standard deviation is taken from the sample rather than from the population.
t-score formula is t = (x-m)/s
t is the t-score
x is the maximum and minimum sample mean at 95% two tail confidence interval.
m is the given sample mean.
s is the standard error.
from the provided t-score table you should be able to derive that the critical t-score with 44 degrees of freedom at 95% two tail confidence interval is plus or minus 2.015.
the table just shows 2.015.
you have to extrapolate from that to determine that you need 2.015 on the high end of the confidence interval and -2.015 on the low end of the confidence interval.
on the low end of the confidence interval, the t-score formula becomes:
-2.015 = (x - 426) / 28.3235.
solve for x to get x = -2.015 * 28.3235 + 426 = 368.93.
on the high end of the confidence interval, the t-score formula becomes:
2.015 = (x - 426) / 28.3235.
solve for x to get x = 2.015 * 28.3235 + 426 = 483.07
your two tail 95% confidence interval is 368.93 to 483.07.
that should be your answer.
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