Estimates and Sample Sizes — practice test 7B

Mar 31, 2017

Problems with solutions in Elementary Statistics — Estimating a Population Proportion, Estimating a Population Mean: σ Known, Estimating a Population Mean: σ Not Known, Estimating a Population Variance.

Directions:
Choose the correct answer for multiple-choice items.

Solve the problem.
- 1. Find the critical value of z_α_/2that corresponds to a degree of confidence of 98%.
  1. 2.05
  2. 1.75
  3. 2.33
  4. 2.575
- 2. The following confidence interval is obtained for a population proportion, p:
  (0.870, 0.897)
  Use these confidence interval limit to find the point estimate, p̂.
  1. 0.882
  2. 0.885
  3. 0.870
  4. 0.894
Find the margin of error for the 95% confidence interval used to estimate the population proportion.
- 3. In a clinical test with 2440 subjects, 70% showed improvement from the treatment.
  1. 0.0196
  2. 0.0236
  3. 0.0182
  4. 0.0175
Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
- 4. n = 107, x = 66; 88 percent
  1. 0.539 < p < 0.695
  2. 0.540 < p < 0.694
  3. 0.543 < p < 0.691
  4. 0.544 < p < 0.690
Find the minimum sample size you should use to assure that your estimate of p̂ will be within the required margin of error the population p.
- 5. Margin of error: 0.07; confidence level: 97%, p̂ and q̂ are unknown.
  1. 240
  2. 241
  3. 111
  4. 112
- 6. Margin of error: 0.008; confidence level: 99%; from a prior study, p̂ is estimated by 0.164.
  1. 12,785
  2. 8230
  3. 14,205
  4. 1145
Solve the problem.
- 7. 38 randomly picked people were asked if they rented or owned their own home, 11 said they rented. Obtain a point estimate of the true proportion of home owners.
  1. 0.737
  2. 0.289
  3. 0.711
  4. 0.224
Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
- 8. A survey of 300 union members in New York State reveals that 112 favor the Republican candidate for governor. Construct the 98% confidence interval for the true population proportion of all New York State union members who favor the Republican candidate.
  1. 0.308 < p < 0.438
  2. 0.304 < p < 0.442
  3. 0.316 < p < 0.430
  4. 0.301 < p < 0.445
Solve the problem.
- 9. A one-sided confidence interval for p can be written as p < p̂ + E or p > p̂ - E where the margin of error E is modified by replacing z_α_/2withz_α. If a teacher wants to report that the fail rate on a test is at most x with 90% confidence, construct the appropriate one-sided confidence interval. Assume that a simple random sample of 58 students results in 6 who fail the test.
  1. p > 0.155
  2. p > 0.052
  3. p < 0.052
  4. p < 0.155
Determine whether the given conditions justify using the margin of error E = z_α_/2 σ/ √n when finding a confidence interval estimate of the population mean μ.
- 10. The sample size is n = 8, and σ is not known.
  1. Yes
  2. No
Use the confidence level and sample data to find the margin of error E.
- 11. College students’ annual earnings: 99% confidence; n = 66, x̅ = $3903, σ = $801.
  1. $254
  2. $8
  3. $230
  4. $1237
Use the confidence level and sample data to find a confidence interval for estimation the population μ.
- 12. A random sample of 100 full-grown lobsters had a mean weight of 22 ounces and a standard deviation of 3.7 ounces. Construct a 98 percent confidence interval for the population mean μ.
  1. 22 < μ < 24
  2. 21 < μ < 23
  3. 20 < μ < 22
  4. 21 < μ < 24
Use the margin of error, confidence level, and standard deviation σ to find the minimum sample size required to estimate an unknown population mean μ.
- 13. Margin of error: $136, confidence level: 99%, σ = $503.
  1. 91
  2. 53
  3. 10
  4. 46
Do one of the following, as appropriate:
(a) Find the critical value z_α_/2;
(b) Find the critical value t_α_/2;
(c) State that neither the normal nor the t distribution applies.
- 14. 90%; n = 9; σ = 4.2; population appears to be very skewed.
  1. z_α_/2 = 2.896
  2. Neither the normal nor the t distribution applies.
  3. z_α_/2 = 2.306
  4. z_α_/2= 2.365
Find the margin of error.
- 15. 95% confidence interval; n = 12; x̅ = 35.6; s = 6.4.
  1. 4.066
  2. 3.659
  3. 3.050
  4. 4.879
Use the given degree of confidence and sample data to construct a confidence interval for the population mean μ. Assume that the population has a normal distribution.
- 16. The amount (in ounces) of juice in eight randomly selected juice bottles are:
  15.1 15.6 15.6 15.9
  15.9 15.2 15.1 15.2
  Construct a 98 percent confidence interval for the mean amount of juice in all such bottles.
  1. 15.77 < μ < 15.13
  2. 15.09 < μ < 15.81
  3. 15.03 < μ < 15.87
  4. 15.87 < μ < 15.03
Solve the problem.
- 17. Find the critical value X²_L corresponding to a sample size of 3 and a confidence level of 90 percent.
  1. 5.991
  2. 9.21
  3. 0.103
  4. 0.0201
Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation σ. Assume that the population has a normal distribution.
- 18. Weights of men: 90% confidence; n = 14, x̅ = 156.9 lb, s = 11.1 lb.
  1. 9.0 lb < σ < 2.7 lb
  2. 8.5 lb < σ < 16.5 lb
  3. 8.7 lb < σ < 14.3 lb
  4. 8.2 lb < σ < 15.6 lb
Find the appropriate minimum sample size.
- 19. To be able to say with 95% confidence level that the standard deviation of a data set is within 10% of the population’s standard deviation, the number of observations within the data set must be greater than or equal to what quantity?
  1. 191
  2. 805
  3. 335
  4. 250
Use the give degree of confidence and sample data to find a confidence interval for the population standard deviation σ. Assume that the population has a normal distribution.
- 20. The amounts (in ounces) of juice in eight randomly selected juice bottles are:
  15.8 15.7 15.0 15.7
  15.2 15.2 15.5 15.0
  Find a 98 percent confidence interval for the population standard deviation σ.
  1. (0.20, 0.78)
  2. (0.20, 0.67)
  3. (0.19, 0.67)
  4. (0.22, 0.84)

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