1. Find the critical value zc necessary to form a confidence interval at the level of confidence shown below.
c = 0.83
2. Use the values on the number line to find the sampling error.
3. Find the margin of error for the given values of c, s, and n.
c = 0.90, s = 3.4, n = 64
4. Match the level of confidence, c = 0.98, with its representation on the number line, given x‾= 59.1, s = 8.9, and n = 55.
5. Construct the confidence interval for the population mean μ.
c = 0.95, x‾= 15.6, s = 3.0, and n = 40
A 95% confidence interval for μ is ( , ).
6. Use the confidence interval to find the margin of error and the sample mean.
(0.374, 0.470)
7. People were polled on how many books they read the previous year. How many subjects are needed to estimate the number of books read the previous year within one book with 99% confidence? Initial survey results indicate that σ = 15.6 books.
8. Find the critical value tc for the confidence level c = 0.50 and sample size n = 17.
t- Distribution table
|
|
Level of
confidence, c
|
0.50
|
0.80
|
0.90
|
0.95
|
0.98
|
0.99
|
|
|
One tail, a
|
0.25
|
0.10
|
0.05
|
0.025
|
0.01
|
0.005
|
|
d.f.
|
Two tails, a
|
0.50
|
0.20
|
0.10
|
0.05
|
0.02
|
0.01
|
|
1
|
|
1.000
|
3.078
|
6.314
|
12.706
|
31.821
|
63.657
|
|
2
|
|
.816
|
1.886
|
2.920
|
4.303
|
6.965
|
9.925
|
|
3
|
|
.765
|
1.638
|
2.353
|
3.182
|
4.541
|
5.841
|
|
4
|
|
.741
|
1.533
|
2.132
|
2.776
|
3.747
|
4.604
|
|
5
|
|
.727
|
1.476
|
2.015
|
2.571
|
3.365
|
4.032
|
|
6
|
|
.718
|
1.440
|
1.943
|
2.447
|
3.143
|
3.707
|
|
7
|
|
.711
|
1.415
|
1.895
|
2.365
|
2.998
|
3.499
|
|
8
|
|
.706
|
1.397
|
1.860
|
2.306
|
2.896
|
3.355
|
|
9
|
|
.703
|
1.383
|
1.833
|
2.262
|
2.821
|
3.250
|
|
10
|
|
.700
|
1.372
|
1.812
|
2.228
|
2.764
|
3.169
|
|
11
|
|
.697
|
1.363
|
1.796
|
2.201
|
2.718
|
3.106
|
|
12
|
|
.695
|
1.356
|
1.782
|
2.179
|
2.681
|
3.055
|
|
13
|
|
.694
|
1.350
|
1.771
|
2.160
|
2.650
|
3.012
|
|
14
|
|
.692
|
1.345
|
1.761
|
2.145
|
2.624
|
2.977
|
|
15
|
|
.691
|
1.341
|
1.753
|
2.131
|
2.602
|
2.947
|
|
16
|
|
.690
|
1.337
|
1.746
|
2.120
|
2.583
|
2.921
|
|
17
|
|
.689
|
1.333
|
1.740
|
2.110
|
2.567
|
2.898
|
|
18
|
|
.688
|
1.330
|
1.734
|
2.101
|
2.553
|
2.878 |
|
19
|
|
.688
|
1.328
|
1.729
|
2.093
|
2.539
|
2.861
|
|
20
|
|
.687
|
1.325
|
1.725
|
2.086
|
2.528
|
2.845
|
|
21
|
|
.686
|
1.323
|
1.721
|
2.080
|
2.518
|
2.831
|
|
22
|
|
.686
|
1.321
|
1.717
|
2.074
|
2.508
|
2.819
|
|
23
|
|
.685
|
1.319
|
1.714
|
2.069
|
2.500
|
2.807
|
|
24
|
|
.685
|
1.318
|
1.711
|
2.064
|
2.492
|
2.797
|
|
25
|
|
.684
|
1.316
|
1.708
|
2.060
|
2.485
|
2.787
|
|
26
|
|
.684
|
1.315
|
1.706
|
2.056
|
2.479
|
2.779
|
|
27
|
|
.684
|
1.314
|
1.703
|
2.052
|
2.473
|
2.771
|
|
28
|
|
.683
|
1.313
|
1.701
|
2.048
|
2.467
|
2.763
|
|
29
|
|
.683
|
1.311
|
1.699
|
2.045
|
2.462
|
2.756
|
|
co
|
|
.674
|
1.282
|
1.645
|
1.960
|
2.326
|
2.576
|
9. Construct the indicated confidence interval for the population mean μ using (a) a t-distribution. (b) If you had incorrectly used a normal distribution, which interval would be wider?
c = 0.99, x‾= 12.2, s = 2.0, and n = 5
a. The 99% confidence interval using a t-distribution is ( , ).
b. If you had incorrectly used a normal distribution, which interval would be wider?
10. In a random sample of five microwave ovens, the mean repair cost was $70.00 and the standard deviation was $12.50. Assume the variable is normally distributed and use a t-distribution to construct a 90% confidence interval for the population mean μ. What is the margin of error of μ?
11. The monthly incomes for 12 randomly selected people, each with a bachelor's degree in economics, are shown below.
Assume the population is normally distributed.
4450.36 4596.43 4366.26
4455.52 4151.31 4407.46
4283.11 4527.74 4407.46
3946.48 4023.48 4221.98
a. Find the sample mean.
b. Find the sample standard deviation.
c. Construct a 99% confidence interval for the population mean μ.
12. In a random sample of 75 bolts, the mean length was 1.77 inches and the standard deviation was 0.08 inch. Use a normal distribution or a t-distribution to construct a 90% confidence interval for the mean.
Which distribution should be used to construct the 90% confidence interval?
13. Let p be the population proportion for the following condition. Find the point estimates for p and q.
In a survey of 1641 adults from country A, 560 said that they were not confident that the food they eat in country A is safe.
14. A researcher wishes to estimate, with 95% confidence, the proportion of adults who have high-speed Internet access. Her estimate must be accurate within 2% of the true proportion.
a. Find the minimum sample size needed, using a prior study that found that 46% of the respondents said they have high-speed Internet access.
b. No preliminary estimate is available. Find the minimum sample size needed.
15. The table to the right shows the results of a survey in which 2564 adults from Country A, 1120 adults from Country B, and 1085 adults from Country C were asked if human activity contributes to global warming. Complete parts (a), (b), and (c).
|
Adults who say that human activity contributes to global warming
|
|
Country A
|
65%
|
|
Country B
|
86%
|
|
Country C
|
93%
|
a. Construct a 99% confidence interval for the proportion of adults from Country A who say human activity contributes to global warming.
b. Construct a 99% confidence interval for the proportion of adults from Country B who say human activity contributes to global warming.
c. Construct a 99% confidence interval for the proportion of adults from Country B who say human activity contributes to global warming.
16. Find the critical values χ2L and χ2R for the given confidence level c and sample size n.
c = 0.95, n = 26
17. You randomly select and measure the contents of 10 bottles of cough syrup. The results (in fluid ounces) are shown.
4.214 4.297 4.252 4.247 4.183
4.288 4.263 4.246 4.228 4.239
Assume the sample is taken from a normally distributed population. Construct 80% confidence intervals for
a. The population variance σ2.
b. The population standard deviation σ.
18. The monthly incomes of 13 randomly selected individuals who have recently graduated with a bachelor's degree in economics have a sample standard deviation of $349. Construct a confidence interval for the population variance σ2 and the population standard deviation σ. Use a 90% level of confidence. Assume the sample is from a normally distributed population.
19. The number of hours of reserve capacity of 10 randomly selected automotive batteries is shown.
1.79 1.86 1.53 1.63 1.78
1.98 1.39 1.53 1.41 2.02
Assume the sample is taken from a normally distributed population. Construct 90% confidence interval for
a. The population variance σ2.
b. The population standard deviation σ.