**STAT20029 (T2,2020) Questions: ****Week 9**

**Refer Chapter 9:****Hypothesis Testing for single populations**

**Practice Problem 9.1**

P9.1(a): For each of the following, state the null and alternative hypotheses.

(a) The mean length increased above 15cm.

(d) The proportion of healthy children in a community is believed to be less 75%.

(e) The average weight of people in a community is 85kg.

**Practice Problem 9.2**

9.2(a): For each of the hypothesis statements in problem 9.1, is a two-tailed test or one tailed test needed? If it is a one-tailed test, indicate if it is a left-tailed or right tailed test.

(a) The mean length increased above 15cm.

(d) The proportion of healthy children in a community is believed to be less 75%.

(e) The average weight of people in a community is 85kg.

**Practice Problem 9.5**

P9.5: From the information given, indicate if a correct decision, a Type I error or a Type II error was made.

(a) H_{0}: µ = 1.5 litres. The decision was to not reject H0 and µ is actually 1.5 litres.

(b) H_{0}: µ = 1.5 litres. The decision was to not reject H0 and µ is actually 1.6 litres.

(c) H_{0}: µ = 1.5 litres. The decision was to reject H0 and µ is actually 1.5 litres.

**Activity 9.1**

Activity 9-1: Find the critical Z - value(s) for each situation and draw the appropriate figure.

(a) A left tail test with α = 0.01

(b) A two- tail test with α = 0.20

(c) A two- tail test with α = 0.05 with rejection region and non-rejection region

**Practice Problem 9.4**

P9-4: The critical value for a specific left-tailed hypothesis test is given as –1.645. Depending on the sample selected for the hypothesis test, the calculated test statistic can vary. For each of the following test statistics, indicate whether to reject the null hypothesis.

(a) –2.8

(b) –1.7

(c) 0

**Practice Problem 9.8**

PP9.8(a): Use the data given to test the following hypotheses.

**Practice Problem 9.6(a)**

PP9.6(a): Use the data given to test the following hypotheses.

**Practice Problem 9.7**

PP9.7: Use the data given to test the following hypotheses.

**Practice Problem 9.7** (**P-value approach)**

PP9.7: Use the data given to test the following hypotheses.

**Practice Problem 9.8(b)**

PP9.8(b): Use the data given to test the following hypotheses. Use the p-value to draw a statistical conclusion.

**Practice Problem 9.6(b)**

**PP9.6(b): ****Use the data given to test the following hypotheses. Use the p-value to reach a statistical conclusion.**

**Activity 9.2**

Activity 9.2: A survey claims that 9 out of 10 doctors recommend Panadol for their patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend Panadol is less than 0.90, a random sample of 60 doctors was selected. Suppose the test statistic is -2.30. Can we conclude that Ho should be rejected at the

(A) α = 0.10, (B) α = 0.05, and (C) α = 0.01 level of Type I error. Give answer for each case separately.

**Activity 9.3**

Activity 9.3: Should we reject or not reject a null hypothesis if the p-value is lower than the critical value?

**Practice Problem 9.6(c) CI method**

PP9.6: (c) Use the data given to test the following hypotheses. Use the confidence interval approach to reach a statistical conclusion.

**Review Problem 9.2**

RP9.2: Use the information given and the eight-step approach to test the following hypotheses. Let α = 0.05. Assume the population is normally distributed.

**Practice Problem 9.18**

PP9.18: A hole-punch machine is set to punch a hole 1.84 cm in diameter in a strip of sheet metal in a manufacturing process. The strip of metal is then creased and sent on to the next phase of production, where a metal rod is slipped through the hole. It is important that the hole be punched to the specified diameter of 1.84 cm. To test punching accuracy, technicians have randomly sampled 12 punched holes and measured the diameters. The data follow. Use α = .10 to determine whether the holes are being punched with an average diameter of 1.84 cm. Assume that the diameters of the punched holes are normally distributed in the population.

**Practice Problem 9.21**

PP9.21: According to a survey, the average commuting time from home to the city is 19.0 minutes if the population is between 1 and 3 million people. A researcher lives in a city with a population of 2.4 million people and wants to test the claim. A random sample of commuters is gathered, and the data are collected and analysed using Excel. The Excel output is shown here. Using α = .05, what statistical conclusion can be reached?

**Practice Problem 9.22**

P9.22: You are testing H_{0 }: *p* ≥ 0.25 versus H_{a} : *p *< 0.25. A random sample of 146 people gives a value of p_{s} = 0.21. Using α = 0.10, test this hypothesis.

**Review Problem 9.10**

RP9.47: Changes in lifestyles over the years are believed to now see families eat more takeaway food at home for dinner than previously. It is claimed the proportion of families who now eat at least one takeaway meal per week as the family dinner is 75%. To test this claim, a random sample of 250 families was selected. 192 families indicated that they eat at least one takeaway meal per week as the family dinner. Using a 5% level of significance, what conclusion can be made? If the actual proportion of families who eat at least one takeaway meal per week as the family dinner was later found to be 83%, what error if any would have been made?

**Refer Chapter 10: Statistical inferences about two populations**

**Activity 9.4**

Activity 9-4: A survey found that the average hotel room rate in Sydney is $88.42 and the average room rate in Brisbane is $80.61 and that the population standard deviations were $5.62 and $4.83 respectively. Assume that the data were obtained from two samples of 50 hotels each. At α = 0.05, can it be concluded that there is no significant difference in the rates?

**Practice Problem 10.7**

PP10-7: A company’s auditor believes the per diem cost in Darwin rose significantly between 2005 and 2012. To test this belief, the auditor samples 51 business trips from the company’s records for 2005; the sample average was $190 per day, with a population standard deviation of $18.50. The auditor selects a second random sample of 47 business trips from the company’s records for 2012; the sample average was $198 per day, with a population standard deviation of $15.60. If he uses a risk of committing a Type I error of .01, does the auditor find that the per diem average expense in Darwin has gone up significantly?

**Activity 9.5**

Activity 9-5: A maintenance supervisor is comparing the standard version of an instructional booklet with one that has been claimed to be superior. An experiment is conducted in which 26 technicians are divided into two groups, provided with one of the booklets, then given a test a week later. For the 13 using the standard version, the average exam score was 72.0, with a standard deviation of 9.3.For the 13 given the new version, the average score was 80.2, with a standard deviation of 10.1. Assuming normal populations with variances equal and using the 0.05 level of significance, does the new booklet to be better than the standard version.

**Practice Problem 10.15**

PP10-15: Based on an indication that mean daily car-rental rates may be higher for Melbourne than for Sydney, a survey of eight car-rental companies in Melbourne is taken and the sample mean car-rental rate is $47, with a standard deviation of $3. Further, suppose a survey of nine car-rental companies in Sydney results in a sample mean of $44 and a standard deviation of $3. Use α = .01 to test whether the average daily car-rental rates in Melbourne are significantly higher than those in Sydney. Assume car-rental rates are normally distributed and the population variances are equal.

**Practice Problem 10.30**

PP10.30**: **According to a study conducted for Gateway Computers, 59% of men and 70% of women say that weight is an extremely/very important factor in purchasing a laptop computer. Suppose this survey was conducted using 374 men and 481 women. Do these data show enough evidence to declare that a significantly higher proportion of women than men believe that weight is an extremely/very important factor in purchasing a laptop computer? Use a 5% level of significance.

**Practice Problem 10.33**

PP10-33: According to the Australian Government, in 2014 there were 485 job service providers listed in Victoria, of whom 41 provide job training, and in Queensland there are 382 listed job service providers of whom 62 provide job training. Test at 1% level of significance whether the difference in proportions of job service providers who offer training between these two states is significant.

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