Chapter 7

Practice Exercise

*** Answer the followings

7.1 Assume that the distribution of a population is a normal distribution with the standard deviation of 50. The sample mean of 25 samples randomly selected from the population is 70. Test the hypothesis \(\small H_0 : \mu = 100 , H_1 : \mu \ne 100 \) with the significance level of 0.01 and obtain the \(p\)-value.

7.2 A bolt manufacturer claims that the average of bolt length is 4.5 cm, the standard deviation is 0.020 cm and it is normally distributed. When 16 samples were taken randomly, the sample mean was 4.512. Can you say that the actual average length of the bolt is different from that of the manufacturer’s claim? Assume that the significance level is 0.01.

7.3 The amount of water needed in the process of producing a compound with distilled water added to any component in a fixed amount depends on the purity of the component. In the manufacturer's experience, the water requirement for a normal production is 6 liters and the standard deviation is 1 liter. A sample mean from samples of nine products was 7 liters. Can products be considered as a normal production at the significance level of 0.05?

7.4 A psychologist is working on physically disabled workers. Based on the past experience, the psychologist believed that the average social (relationship) score of these disabled workers was greater than 80. Twenty employees were sampled from the score population to obtain the following result:

99, 69, 91, 97, 70, 99, 72, 74, 74, 76,

96, 97, 68, 71, 99, 78, 76, 78, 83, 66.

The psychologist wants to know if the average social score of the population is right. Assume that the population follows a normal distribution and its standard deviation is 10. Test with the significance level of 0.05.

7.5 The following is the weights of the 10 employees randomly selected who are working in the shipping department of a wholesale food company.

154, 154, 186, 243, 159, 174, 183, 163, 192, 181 (unit pound)

Based on this data, can you say that the average weight of employees working in the shipping department is greater than 160 pound? Use the significance level of 5%.

7.6 A company that makes tar on a roof wants an average of less than 3 percent of impurities. By sampling 30 barrels of tar, can this data suggest that the population mean is less than 3% when the proportion of impurities is as follows? Use the significance level of 5%.

3 3 1 1 0.5 2 2 4 5 4 5 3 1 3 1

4 1 1 4 2 5 3 1 1 1 0.75 1.5 3 3 2

7.7 In a large manufacturer, the manager of the company claims that the average adaptation score of all unskilled workers is greater than 60. In order to check this claim, 40 unskilled workers were randomly selected and their test scores of adaptation scores were as follows.

73 57 96 78 74 42 55 44 91 91 50 65 46 63 82 60 97 79 85 79

92 50 42 46 86 81 81 83 64 76 40 57 78 66 84 96 94 70 70 81

If the population variance is 280, test the hypothesis at the significance level of 0.05 whether the manager's argument is correct. What is the \(p\)-value?

7.8 Existing tires of a company have an average life span of 54,000 km and a standard deviation of 10,000 km. Two researchers conducted a performance test for new tires independently. The first researcher tested 25 tires and obtained an average life span of 48,000 km, while the second researcher tested 100 to get 50,000 km. Which test result gives more reliable statistical evidence that the average life span of a new tire is less than the existing one? Assume that the standard deviation has not changed.

7.9 Suppose \(n\) = 100 and \(\sigma\) = 8.4 are given to test the hypothesis \(\small H_0 : \mu = 75.0 , H_1 : \mu \lt 75.0 \).

1) If the null hypothesis is rejected when the sample mean is less than 73.0, what is the probability of the type 1 error?

2) If we test the alternative hypothesis \(\small H_1 : \mu \gt 75 \) and the decision rule is the same as above, what is the probability of the type 1 error?

7.10 A random variable follows a normal distribution N(μ, 4). Test the hypothesis \(\small H_0 : \mu = 0 \) with the significance level of 0.10 if \(n\) = 25 and \(\small \overline x\) = 0.28. Does the 90% confidence interval include μ = 0?

7.11 A sample of size 21 was randomly taken from a population that follows the normal distribution and its sample variance was 10. Test the null hypothesis \(\small H_0 : \sigma^2 \) = 15 and the alternative hypothesis \(\small H_1 : \sigma^2 \) ≠ 15 with the significance level of 0.05.

7.12 If the variance of diameters of metal washer products is less than \( 0.000005^2\), then the production process is under control. 31 samples were randomly selected from the assembly line and its variance is \( 0.000006^2\). According to this data, is the assembly process out of control with the significance level of 0.05? What assumptions do you need to get an answer?

7.13 For a manufacturer to make a product, the variance of tensile strength of the synthetic fiber must be less than or equal to five. When 25 samples are randomly selected from the new shipment and the variance is seven. Does this data provide sufficient evidence for the manufacturer to reject the shipment? Assume the significance level of 0.05 and that the tensile strength of the fiber follows approximately a normal distribution.

7.14 In a process of filling the container, the average weight is set to 8g and the variance of the weight shall be \(\sigma^2\) = 4g to satisfy the given tolerances. 25 container samples were randomly selected and its standard deviation was 2.8g.

1) If the weight is assumed to follow a normal distribution, is the population variance greater than the prescribed value \(\sigma^2\)? Test at the significance level of 0.01.

2) What is the range of sample variances that can not be rejected? Are these values symmetrical to the prescribed value of 2g? Why is that?

7.15 A university wants to build a student parking lot. School authorities think more than 20 percent of students go to school by car. 250 students are randomly selected and 65 of them said that they go to school by car. Test at the significance level of 0.05 whether the school authorities' thinking is correct.

7.16 The accountant of a company thinks that more than 20% of the statement of expenses includes at least one mistake. 400 statements of expenses were randomly selected and found at least one mistake in 100 statements. Test whether the accountant's belief is correct with the significance level of 0.05.

7.17 A researcher met 200 office workers of a company who changed their job last year. Thirty of them stated that they changed their work, because they could not expect for promotion. Can you say that less than 20% of the employees were changed their job, because of their promotion? The significance level is 0.05.

7.18 A statistician threw coins 24,000 times, with 12012 front and 11988 back. Does this data support the null hypothesis that there is 0.5 chance that the front face will appear? Test at the significance level of 0.05.

7.19 A high school student made dice from wood. It doesn't look like a cube, so it's not one-sixth of a chance of six at the top. I rolled the dice 18,000 times and got 3,126 of six at the top.

1) Determine whether these 3126 occurrences are statistically significant by obtaining the \(p-value\).

2) Since it is too much to demand that a hand-made dice have an exact one-sixth chance of producing six, we decided to allow the error of 0.01 chance to be a fair dice. Using the above sample experiment, obtain a confidence interval for the probability of a 6 and determine whether it is acceptable to be viewed as a fair dice (Note: It is difficult to distinguish between the statistical significance and the practical significance by relying solely on testing hypothesis.)

** Obtain a power function for each of the following situations and draw a graph:

7.20 \(\small H_0 : \mu = 51 , H_1 : \mu < 51 , n = 25, \sigma = 3 , \alpha = 0.05 \).

7.21 \(\small H_0 : \mu = 516 , H_1 : \mu > 516 , n = 16, \sigma = 32 , \alpha = 0.05 \).

7.22 \(\small H_0 : \mu = 3 , H_1 : \mu \ne 3 , n = 100, \sigma = 1 , \alpha = 0.05 \).

7.23 \(\small H_0 : \mu = 4.25 , H_1 : \mu > 4.25 , n = 81, \sigma = 1.8 , \alpha = 0.01 \).

7.24 On question 7.21, find \(n, C\), and the decision rule when β = 0.10 and \(\mu_1\) = 520.

7.25 On question 7.23, find \(n, C\), and the decision rule when β = 0.05 and \(\mu_1\) = 4.52.

7.26 On question 7.23, find \(n, C\), and the decision rule when β = 0.03 and \(\mu_1\) = 5.

7.27 The standard deviation of the resistance of a wire made by a factory is known to be 0.02 ohms. An electronics company has decided not to buy the wire if there is sufficient evidence that the average resistance of the wire is greater than 0.4 ohms. The company's management adheres to the policy of the significance level α = 0.05 and the sample size \(n\) = 100 in statistical tests.

1) Describe the proper null and alternative hypothesis.

2) Describe the type 1 and type 2 errors.

3) Obtain the rejection region of the test.

4) Draw the power function and operating characteristic curve.

7.28 In order to test whether the coins were fair, the following decision criteria were established:

① If the number of fronts when a coin thrown 100 times is in between 40 and 60, the null hypothesis is adopted.

② In other cases, the null hypothesis is rejected.

1) What is the null and alternative hypothesis?

2) What is the probability of the type 1 error in this test?

3) Mark the critical value of this test using the standard normal distribution.

4) Calculate the probability of the type 2 error when \(p\) = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9.

7.29 The strength of glass produced by a glass company is said to be normally distributed and the variance is 9200. If the strength of the glass is less than 2500 psi, it cannot be sold on the market and is judged as defective. That is, the lower specification limit is \(L\) = 2500psi. If the defective rate of a lot containing a lot of glass is 1%, the lot is accepted, and if the defective rate is 5%, it is rejected. What should be the sample size and decision criteria? However, the decision criteria should reflect the producer risk probability of rejecting a good lot, \(\alpha\) = 1%, and the consumer risk probability of rejecting a bad lot, \(\beta\) = 10%.

7.30 The moisture content of frozen meat produced by any food company must not exceed 20%. The moisture content of all frozen meat is normally distributed and the standard deviation is 0.05%. If a lot of frozen meat has a defective rate of 1% (defective if the moisture content exceeds 20%), the lot is accepted, and if the defective rate is 2.5%, it is rejected. What should be the sample size and decision criteria? The decision criteria should reflect the producer risk probability of rejecting a good lot, \(\alpah\) = 2.5%, and the consumer risk probability of rejecting a bad lot, \(\beta\) = 5%.

Multiple Choice Exercise

Practice Exercise