PBT150S Final tutorial 2019 PDF

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Lecturer: JC Kabala

20 October 2018

Operations Management Techniques 1: PBT150S Final Tutorial – Test 4 Question 1 1.1. A research engineer for a tire manufacturer is investigating tire life for a new rubber compound and has built 16 tires and tested them to end-of-life in a road test. The sample mean and standard deviation are 60,139.7 and 3645.94 kilometres. Find a 95% confidence interval on mean tire life. 1.2. A manufacturer of electronic calculators is interested in estimating the fraction of defective units produced. A random sample of 800 calculators contains 10 defectives. Compute a 89% two-sided confidence interval on the fraction defective. 1.3. A manufacturer of car batteries claims that the batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5, and 4.2 years, construct a 95% confidence interval for 𝜎 2 and decide if the manufacturer’s claim that 𝜎 2 = 1 is valid. Assume the population of battery lives to be approximately normally distributed. Question 2 2.1. The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 13 defectives. Find a 95% two-sided confidence interval on the fraction of defective circuits produced by this particular tool. 2.2. A random sample of 40 chocolate energy bars of a certain brand has, on average, 230 calories per bar, with a standard deviation of 15 calories. Construct a 99% confidence interval for the true mean and standard deviation calorie content of this brand of energy bar. Assume that the distribution of the calorie content is approximately normal. Question 3 3.1. The National Association of Automobile Manufacturers of South Africa reported that automobile plants in the South Africa required an average of 24.9 hours to assemble a new car. In order to reduce inventory costs, a new “jus t-in-time” parts availability has been introduced on the assembly line. Suppose that a random sample of 49 cars showed a sample mean assembly time under the new system was 25.2 hours with sample standard deviation of 1.6 hours. At 5% level of significance, does this information indicate that the population mean assembly time is different under the new system? 3.2. A scrap metal dealer claims that the mean of his cash sales is “no more than $80,” but an Internal Revenue Service agent believes the dealer is untruthful. Observing a sample of 20 cash customers, the agent finds the mean purchase to be $91, with a standard deviation of $21. Assuming the population is approximately normally distributed, and using the 0.05 level of significance, is the agent’s suspicion confirmed? Question 4 4.1. According to the National Association of Home Builders, 55% of new single-family homes built during 2005 had a fireplace. Suppose a nationwide homebuilder has claimed that its homes are “a cross section of America,” but a simple random sample of 600 of its single-family homes built during that year included only 50.0% that had a fireplace. Using the 0.05 level of significance in a two-tail test, examine whether the percentage of sample homes having a fireplace could have differed from 55% simply by chance. 4.2. A weather service claims that the standard deviation of the number of fatalities per year from tornadoes is no more than 25. A random sample of the number of deaths for 28 years has a standard deviation of 31 fatalities. At is there enough evidence to reject the weather service’s claim? Page 1 of 3

Lecturer: JC Kabala

20 October 2018

Question 5 5.1. A nutritionist believes that obesity is more prevalent among American adults than it was in the past. He discovers that in a study conducted in the year 1994, 380 of the 1630 randomly chosen adults were classified as obese. However, in a more recent study, he finds 726 out of 2350 randomly chosen adults were classified as obese. At α = 0.05, do these studies provide evidence to support the nutritionistʹs claim that the proportion of obese adults has significantly increased since 1994? 5.2. A random sample of 21 from population A has mean 56 and standard deviation 12. A random sample of 26 from population B has mean 63 and standard deviation 14. Assuming that the variances of their respective populations are equal, test the claim that the population means are equal at the 5% significance level. Question 6 6.1. A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten subjects are randomly selected and pretested. The results are listed below. The subjects were placed on the drug for a period of 6 months, after which their cholesterol levels were tested again. The results are listed below. (All units are milligrams per deciliter.) Test the company’s claim that the drug lowers cholesterol levels. Use α = 0.01. Assume that the distribution is normally distributed. Subject Before 19 1 22 2 20 3 19 4 17 5 6 25 23 7 8 26 19 9 10 24

After 18 22 21 17 17 25 20 25 19 22

6.2. Consumer testing service compared gas ovens to electric ovens by baking one type of bread in five ovens of each type. Assume the baking times are normally distributed. The gas ovens had an average baking time of 0.9 hours with standard deviation of 0.09 hours and the electric ovens had an average baking time of 0.7 hours with standard deviation of 0.16 hours. Test the hypothesis of identical mean baking times for the two kinds of ovens at the 5% level of significance. Assume unequal variances. Question 7 Upon leaving an assembly area, production items are examined and some of them are found to be in need of either further work or total scrapping. Tags on a sample of 150 items that failed final inspection show both the recommended action and the identity of the inspector who examined the item. The summary information for the sample is shown below. Recommended Action Major Rework Minor rework scrap

Inspector A B C 20 14 13 18 16 23 16 21 9

At the 0.10 level, is the recommended action independent of the inspector?

Page 2 of 3

Lecturer: JC Kabala

20 October 2018

Question 8 Three types of medium sized cars assembled in South Africa have been test driven by a motoring magazine and compared on a variety of criteria. In the area of fuel efficiency performance, five cars of each brand were each test driven 1000 km; the km per litre data are obtained as follows: Brand A 7.60 8.40 8.00 7.60 8.40

Brand B 7.80 8.00 9.10 8.50 9.60

Brand C 9.60 10.40 9.20 9.70 10.60

Count Sum Mean Variance The task is to use One-way ANOVA and to determine at 5% level of significance if there is a difference in the mean fuel consumption for the three makes of car. 𝐧𝐢

(𝐱 𝐢 − 𝐱)𝟐

𝐱 𝐢

𝐧𝐢 ∗ (𝐱 𝐢 − 𝐱)𝟐

𝐧𝐢

Sum ANOVA Source of Variation Between Groups Within Groups

𝐧𝐢 − 𝟏

𝐒𝐢𝟐

(𝐧𝐢 − 𝟏) ∗ 𝐒𝟐𝐢

Sum

SS

df

MS

Fstat

P-value

Fcrit

Total Complete the above tables and answer the following questions (Round all your answers to three decimal places): a) b) c) d) e)

State H0 and Ha and identify the claim Find the critical value(s) and identify the rejection region(s), Find the test statistic, Decide whether to reject or fail to reject the null hypothesis, and Interpret the decision in the context of the original claim.

Question 9 A plastics company that produces automobile dashboard inserts has just received a new injection mould that is supposedly more consistent than the company’s current mould. A quality technician wishes to test whether this new mould will produce inserts that are less variable in diameter than those produced with the company’s current mould. The table shows independent random samples (of size 12) of insert diameters (in centimetres) for both the current and new moulds. New

9.611 9.618 9.594 9.580 9.611 9.597 9.638 9.568 9.605 9.603 9.647 9.590

Current

9.571 9.642 9.650 9.651 9.596 9.636 9.570 9.537 9.641 9.625 9.626 9.579

At α = 0.05 test the claim that the new mould produces inserts that are less variable in diameter than the inserts the current mould produces. Page 3 of 3...