# What is the β-error for the test in part (a) if the true mean life is 41.5 hours? What sample size would be required to ensure that βdoes not exceed 0.05 if the true mean life is 43.5?

Homework #10 Problems

Chapter 9 Problems: 9.2: 3, 6, 7; 9.3: 1, 3, 4, 6; 9.4: 2, 5; 9.5: 3

9.2.3) Output from a software package follows

One-Sample Z:

Test of μ = 35 versus μ > 35

Assumed standard deviation is 1.7

Variable n Sample

Mean

Sample Std.

Deviation

Std Error

(SE) of

Mean

Z0 P-Value

x 25 35.710 0.295

(a) Fill in the missing items. What conclusions would you draw?

(b) Is this a one-sided or a two-sided test?

(c) Use the normal table and the preceding data to construct appropriate one-sided 95% CI on the mean.

(d) What would be the P-value if the alternative is H1 ≠ 35?

9.2.6) A manufacturer produces crankshaft for an automobile engine. The crankshafts wear after 100,000 mile (0.0001 inch) is of interest because it is likely to have an impact on warranty claims. A random sample of n = 12 shafts is tested and 𝑥̅ = 2.75. It is known that σ= 0.6 and that wear is normally distributed.

(a) Test H0: μ= 3 versus H1: μ ≠ 3 using α= 0.05.

(b) What is the power of the test if μ= 3.15?

(c) What sample size would be required to detect a true mean of 3.5 if we wanted the power to be at least 0.95?

9.2.7) The life in hours of a battery is known to be approximately normally distributed with standard deviation σ = 1.0 hours. A random sample of 12 batteries has a mean life of 𝑥̅ = 41 hours.

(a) Is there evidence to support the claim that battery life exceeds 40.5 hours? Use α= 0.05.

(b) What is the P-value for the test in part (a)?

(c) What is the β-error for the test in part (a) if the true mean life is 41.5 hours?

(d) What sample size would be required to ensure that βdoes not exceed 0.05 if the true mean life is 43.5?

(e) Explain how you could answer the question in part (a) by calculating an appropriate confidence bound on life.

9.3.1) A hypothesis will be used to test that a population mean equals 15 against the alternative that the population mean is less than 15 with unknown variance. What is the critical value for the test statistic T0 for the following significance levels?

(a) α = 0.05 and n = 20 (b) α = 0.01 and n = 15 (c) α = 0.1 and n = 18.

9.3.3) For the hypothesis test H0: μ= 10 versus H1: μ ≠ 10 with variance unknown and n = 18, approximate the P-value for each of the following:

(a) t0 = 2.25 (b) t0 = -1.98 (c) t0 = 0.6

9.3.4) Consider the following computer output:

One Sample T:

Test of H0: μ = 12 versus H1: μ ≠ 12

Variable N Sample

Mean

Sample Std.

Deviation

Std Error

(SE) of

Mean

T0 P-value

x 15 12.564 0.0936034 ? ? ?

(a) How many degrees of freedom are there on the t-test statistic?

(b) Fill in the missing values. You may calculate the bounds on the P-value. What conclusions would you draw?

(c) Is this a one-sided or a two-sided test?

(d) Construct a 95% two-sided CI on the mean.

(e) If the hypothesis had been H0: μ = 12 versus H1: μ > 12, would your conclusion change?

(f) If the hypothesis had been H0: μ = 11 versus H1: μ ≠ 11, would your conclusion change? Answer this question by using the CI computed in part (d).

9.3.6) A study of thermal inertia properties of autoclaved aerated concrete used as a building material.

Five samples of the material were tested in a structure, and the average interior temperature (°C) reported were as follows: 23.0, 22.80, 22.10, 22.58, and 22.67.

(a) Test the hypotheses H0: μ = 22.7 versus H1: μ ≠ 22.7, using α = 0.05. Find the P-value.

(b) Check the assumption that interior temperature is normally distributed.

(c) Explain how the question in part (a) could be answered by constructing a two-sided confidence interval on the mean interior temperature.

9.4.2) Consider the test of H0: σ2 = 7 versus H1: σ2 ≠ 7. Approximate the P-value for each of the following test statistics.

(a) 𝜒0

2 = 24.8 𝑎𝑛𝑑 𝑛 = 18 (b) 𝜒0

2 = 12.8 𝑎𝑛𝑑 𝑛 = 10 (c) 𝜒0

2 = 6.4 𝑎𝑛𝑑 𝑛 = 16

9.4.5) The sample standard deviation for a tire life data was 3600.90 kilometers and n = 15.

(a) Can you conclude using α = 0.05, that the standard deviation of tire life is less than 3950 kilometers? State any necessary assumptions about the underlying distribution of the data. Find the P-value for this test.

(b) Explain how you could answer the question in part (a) by constructing a 95% one-sided confidence interval on σ. 9.5.3) The advertised claim for batteries for cell phones is set at 48 operating hours with proper charging procedures. A study of 5500 batteries is carried out and 20 stop operating prior to 48 hours. Do these experimental results support the claim that less than 0.25 percent of the company’s batteries will fail during the advertised time period, with proper charging procedures?

(a) Use a hypothesis-testing procedure with α = 0.05. What is the P-value?

(b) Explain how you could answer the question in Part (a) by constructing a 95% one-sided confidence interval on

What is the β-error for the test in part (a) if the true mean life is 41.5 hours? What sample size would be required to ensure that βdoes not exceed 0.05 if the true mean life is 43.5?