Text book answers unit 11.doc

Statistical Methods and Calculation Skills

Test yourself 111. Frequent checks were made of the spending patterns of tourists returning from

countries in Asia. Results indicated that travelers spent an average of R1 010 per day. In order to determine whether there has been a change in the average amount spent, a sample of 70 travelers was selected and the mean was determined as R1 090 per day with a standard deviation of R300. Is there evidence of a significant increase in the mean amount spent per day at the 0.01 level of significance? Ho: = 1010

Ha: > 1010 = 1%Reject Ho if the calculated z > 2.33

Accept Ho: No evidence of a significant increase in the mean amount spent per day.

2. The desired percentage of silicon dioxide in a certain type of cement is 5.0. A random sample of 36 specimens gave a sample average percentage of 5.21 and a sample standard deviation of 0.38. Use a significant level of 0.01 and test whether the sample result indicates a change in the average percentage.HO : µ = 5

HA : 5 (no indication of ‘>’ or ‘<’)

= 0.01

The alternative hypothesis is , indicating a two-tail test.

The Central Limit Theorem applies therefore we use z-distribution

Reject HO if z-test > 2.58 or z-test < -2.58

z =

= 5.21 - 5.0

= 3.32

Since 3.32 > 2.58 we reject HO at the 0.01 level of significance.

The sample evidence does suggest that there is a significant change in the average percentage of silicone dioxide in a certain type of cement.

3. A nutritionist claims that the mean tuna consumption by a person is 1.55 kg per year. A sample of 60 people shows that the mean tuna consumption by a person is 1.45 kg

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per year with a standard deviation of 0.51 kg. At = 0.02, can you reject the nutritionist’s claim?

HO : µ = 1.55

HA : 1.55 (no indication of ‘>’ or ‘<’)

= 0.02

The alternative hypothesis is , indicating a two-tail test.


Reject HO if z-test > 2.33 or z-test < -2.33

z =

= 1.45 – 1.55

= -1.52

Since -1.52 is in the acceptance area we do not reject HO at the 0.02 level of significance.

The sample evidence does suggest that there is not a significant change in the average.

4. A machine is set to fire 30 g of dried fruit into a box of cereal moving along the production line. A sample of 36 boxes revealed that the average mass of fruit inserted was 30.3 g with a standard deviation of 0.5 g. Is the increase in the amount of fruit inserted significant at the 0.05 level of significance?

Ho: = 30Ha: > 30 = 5%Reject Ho if the calculated z > 1.64

Reject Ho: The increase in the amount of fruit inserted is significant.

5. A company that makes cola drinks states that the mean caffeine contents per bottle of cola is 40 mg. The quality controller is convinced that it is lower. A sample of 30 bottles of cola has a mean caffeine contents of 39.2 mg with a standard deviation of 7.5 mg. At = 0.01, can the quality controller reject the claim?

HO : µ = 40

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HA : < 40

= 0.01


Reject HO if z-test < -2.33

z =

= 39.2 – 40

= -0.58

Since -0.58 is not < -2.33 we do not reject HO at the 0.01 level of significance.

The sample evidence does suggest that there is not a significant decline in the caffeine contents.

6. Hyperactive children are often disruptive in the typical classroom setting because they find it difficult to remain seated for extended periods of time. The typical number of ‘out-of seat’ behaviors was 12.40 per hour. Treatment was applied to a group of 25 hyperactive children and after treatment the ‘out-of-seat’ behaviors reduced to 11.60 per hour with a standard deviation of 3.5. Using = 0.01, can we conclude that this decline is significant?

Ho: = 12.40Ha: < 12.40 = 1%Reject Ho if the calculated t < -2.492

Accept Ho: The decline in out-of-seat behaviours is not significant.

7. Medical research has shown that repeated wrist extension beyond 20 increases the risk of wrist and hand injuries. In each of 24 randomly selected students in the Information Technology field, the wrist extension was recorded while using a mouse with a proposed new design. The sample mean was found to be 24 with a standard deviation of 5. Test the hypothesis that the mean wrist extension for people using the new mouse design is greater than 20C.

Ho: = 20Ha: > 20 = 5%Reject Ho if the calculated t >1.714

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Reject Ho: The higher wrist extension is significant at the 5% level.

8. You are involved in an environmental awareness program and want to test the claim that the mean waste generated by adults is more than 1.8 kilograms per day. In a random sample of 15 adults, you find that the mean waste generated per person per day is 1.9 kg with a standard deviation of 0.54 kg. At a 5% level of significance, is the claim justified?

Ho: = 1.8Ha: > 1.8 = 5%Reject Ho if the calculated t > 1.761

Do not reject Ho: The higher waste generation is not significant.

9. A random sample of 16 unflavoured ice-cream tubs were selected at random and subjected to chocolate flavoring. The sample mean time required to flavour the ice-cream was 13 minutes with a standard deviation of 2 minutes. Perform a hypothesis test at the 1% level of significance to test that the population mean time required to flavour ice-cream is greater than 10 minutes.

Ho: = 10Ha: > 10 = 1%Reject Ho if the calculated t > 2.602

Reject Ho: The higher time it took to flavour the ice cream is significant at the 1% level.

10. A chicken producer claims that the average mass of a particular group of chickens is 1 kg. Before agreeing to purchase, a customer selected a sample of 25 chickens, which yielded a sample mean of 1.12 kg and standard deviation of 0.1 kg. If the masses can be considered to be normally distributed, should the claim be rejected at the 1% level of significance?

Ho: = 1kgHa: 1kg = 1%

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Reject Ho if the calculated t > 2.797 or t< -2.797

Reject Ho: The average mass of the chickens is significantly different from 1 kg.

11. A personnel manager claims that 60% of all single women hired for secretarial jobs leave to get married within two years. An analysis shows that of a random sample of 120 single women, 64 left to be married. Is this evidence consistent with the company’s claim, at a 1% level of significance?

Ho: = 0.6Ha: 0.6 = 1%Reject Ho if the calculated z > 2.58

Accept Ho: 60% of single women leave to get married within 2 years.

12. A plant is producing large numbers of water testing equipment of which, on average, 2% are defective. In a random sample of 1000, 3% are found to be defective. Does this indicate a significant deterioration in the process? Test at a level of significance of 0.02.

Ho: = 0.02Ha: > 0.02 = 2%Reject Ho if the calculated z > 2.05

Reject Ho: The sample evidence does indicate a significant deterioration in the process.

13. A company manufacturing salad dressings claimed that 85% of households eat salad at least once a week. A nutritionist suspects that the percentage is higher than this. She sampled 200 households and finds that 170 of them eat salad at least once a week. Conduct a test to address the nutritionist’s suspicions. Use = 0.10.

Ho: = 0.85Ha: > 0.85 = 10%Reject Ho if the calculated z > 1.28

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Accept Ho: The sample evidence does indicate that there is no significant increase in the percentage of households that eat salad at least once a week.

14. Club 60 claim that senior citizens participating in some sort of exercise have a blood pressure lower than the average of 160 mmHg. To test this claim, 20 active senior citizens were selected at random and their blood pressure was found to average 151 with a standard deviation of 12. Is Club 60’s claim valid at a 10% level of significance?

Ho: = 160Ha: < 160 = 10%Reject Ho if the calculated t < -1.328

Reject Ho: The claim of a lower blood pressure is valid.

15. A manufacturer claims that his market share is 60%. However a random sample of 500 customers reveals that only 275 are users of his product. Test the claim at the 2% level of significance.

Ho: = 0.6Ha: 0.6 = 2%Reject Ho if the calculated z > 2.33

Do not reject Ho: There is significant evidence to suggest that the market share is 60%.

16. The sales manager wants to determine if the average size of orders received by the company’s eastern branch differ significantly from the average size of orders received from the western branch at a 2% level of significance. A random sample of 90 orders from the eastern branch had a mean value of R131.60 with a standard deviation of R25.80. A random sample of 55 orders received by the western branch had a mean value of R115.70 with a standard deviation of R32.23.

Ho: E = W

Ha: E W = 2%Reject Ho if the calculated z < -2.33 or > 2.33

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Reject Ho: The average size of the orders received by the two branches differs significantly.

17. A large bank is affiliated with both the Mastercard and Visa credit cards. For a sample of 100 Mastercard holders, it is observed that the average month-end account balance is R680 with a standard deviation of R300. For a random sample of 100 Visa cardholders, the average month-end account balance is R550 with a standard deviation of R265. Is the average for the Visa card holders significantly lower than the Mastercard average? Test at = 0.10.

Ho: M = V

Ha: M > V = 10%Reject Ho if the calculated z > 1.28

Reject Ho: There is significant evidence to suggest that the average for the Visa card holders is significantly lower.

18. A consumer testing service compared gas ovens to electric ovens by baking one type of bread in five ovens of each type. The gas ovens had an average baking time of 0.9 hours with a standard deviation of 0.09 hours and the electric ovens had an average baking time of 0.7 hours with a standard deviation of 0.16 hours. Test the hypothesis that the baking times are the same in both kinds of ovens at the 5% level of significance. Assume the baking times are normally distributed.

Ho: G = E

Ha: G E = 5%Reject Ho if the calculated t < -2.306 or > 2.306

Reject Ho: There is significant evidence to suggest that the baking times are not the same in the two kinds of ovens.

19. In order to conduct a consumer behaviour survey, a sample of 500 residents was selected in a metropolitan area. One of the questions asked was “Do you enjoy shopping for clothing?” Of 240 males, 136 answered yes. Of 260 females, 224 answered yes. Determine whether there is evidence that the proportion of females who enjoy shopping for clothing is higher than the proportion of males, using a 5% level of significance.

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Ho: M = E

Ha: M < E

= 5%Reject Ho if the calculated z < -1.64

Reject Ho: The proportion of females who enjoys shopping for clothing is higher than the males’ proportion.

20. In an experiment to compare the fracture toughness of high-purity steel with commercial-purity steel of the same type, 32 specimens were selected from each type. The sample mean and standard deviation toughness for the high-purity steel specimens were 65.6 and 1.4 respectively. The sample mean and standard deviation toughness for the commercial-purity steel specimens were 59.2 and 1.1 respectively. Test at a 5% level of significance whether a significant difference exists between the two types.

Ho: H = C

Ha: H C = 5%Reject Ho if the calculated z > 1.96 or if z < -1.96

Reject Ho: There is significant evidence to suggest that there is a difference between the strength of high-purity steel and commercial-purity steel.

21. A supermarket chain is interested in determining whether a difference exists between the mean shelf life (in days) of two different brands of bread. Random samples of 50 freshly baked loaves of each brand were tested with the results shown below:

Brand A Brand BSample mean 4.1 5.2Sample standard deviation 1.2 1.4

Is there sufficient evidence to conclude that brand B has a longer shelf life than brand A at a 2% level of significance?

Ho: A = B

Ha: A < B = 2%Reject Ho if the calculated z < -2.05

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Reject Ho: The shelf life of Brand B is longer than Brand A.

22. In a public opinion survey, 60 out of a sample of 100 high-income voters and 40 out of a sample of 75 low-income voters supported a decrease in V.A.T. Can we conclude at a 5% level of significance that the proportion of voters favouring a decrease differs between high- and low-income voters?

Ho: H = L

Ha: H L

= 5%Reject Ho if the calculated z < -1.96 or > 1.96

Accept Ho: The proportions favouring a decrease do not differ significantly between high and low income voters.

23. In an AIDS awareness program, it was found that 110 males in a random sample of 310 males are aware of AIDS. In another similar program, it was found that 87 women in a random sample of 290 women are aware of AIDS. Test at the 2% level of significance whether the first campaign was more successful.

Ho: M = F

Ha: M > F

= 2%Reject Ho if the calculated z > 2.05

Accept Ho: The first campaign was not more successful.

24. Tests have been carried out on the effects of three fertilizers on sugar cane growth. Each fertilizer was tried on several different plots of land. Each value is a number of plots of land.

FertilizerA B C

Strong growth 94 124 44Weak growth 50 96 38

Test for an association between the choice of fertilizer and plant growth at a 1% level.

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Ho: No association between choice of fertilizer and plant growthHa: Is an association between choice of fertilizer and plant growth = 1%df = (2-1)(3-1) = 2Reject Ho if the calculated > 9.210

94 124 44 26250 96 38 184144 220 82 446

fo fe

94 84.59 1.0550 59.41 1.49124 129.24 0.2196 90.76 0.0644 48.17 0.3638 33.83 0.51446 446 3.68

Accept HO: No association between choice of fertilizer and plant growth.

25. A car manufacturer is interested in predicting purchase patterns for a new small capacity car they are producing. The car comes in four colours and the manufacturer wants to relate colour preference to the gender of the purchaser. Use the following sample data and do the hypotheses test at a 10% level of significance.

White Green Red SilverMale 260 240 175 420Female 130 200 240 340

Ho: No purchase pattern between colour and genderHa: There is a purchase pattern between colour and gender = 10%df = (4-1)(2-1) = 3Reject Ho if the calculated > 6.251

260 240 175 420 1095130 200 240 340 910390 440 415 760 2005

fo fe

260 212.99 10.35130 177.01 12.48240 240.30 0.0004

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200 199.70 0.0005175 226.65 11.77240 188.35 14.16420 415.06 0.06340 344.94 0.072005 2005 48.89

Reject HO: There is a purchase pattern between colour and gender.

26. Two different manufacturers supply parts for a production process. Each part is tested for six possible defects. The following table shows the number of each type of defect by each supplier. Would you conclude that the defect is independent of the supplier, using a 2.5% level of significance?

DefectSupplier 1 2 3 4 5 6

A 35 10 10 2 5 10B 45 20 0 10 15 20

Ho: Defect is independent of supplierHa: Defect depends on supplier = 2.5%df = (6-1)(2-1) = 5Reject Ho if the calculated > 12.833

35 10 10 2 5 10 7245 20 0 10 15 20 11080 30 10 12 20 30 182

fo fe

35 31.65 0.3545 48.35 0.2310 11.87 0.2920 18.13 0.1910 3.96 9.210 6.04 6.042 4.75 1.5910 7.25 1.045 7.91 1.0715 12.09 0.7010 11.87 0.2920 18.13 0.19182 182 22.26

Reject HO: Defect depends on supplier.

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27. A sales manager has become interested in the number of sales calls made by each of the employees. He reasoned that if all the employees are working equally hard, they should make the same number of calls during a set period of time. In order to investigate this hypothesis, the manager uses a sample of 5 employees and recorded the number of calls they made during a set time period. At a 1% level of significance, is the manager’s idea supported?

Employee A B C D ENo. of calls 31 62 59 40 58

Ho: Employees make the same number of callsHa: Employees do not make the same number of calls = 1%df = 5-1 = 4Reject Ho if the calculated > 13.28

fo fe

31 50 7.2262 50 2.8859 50 1.6240 50 2.0058 50 1.28250 250 15.00

Reject HO: The number of calls made by the employees differs.

28. An accountant for a department store knows from past experience that 23% of the customers pay cash for their purchases, 35% write cheques and the remaining 42% use credit cards. A random sample of 200 sales receipts during a month-end week was examined and the following results were obtained:

Cash Cheque Credit card TotalNumber of customers 37 47 116 200

Is the customers’ payment methods still the same as before? Use = 0.05.

Ho: Customers’ payment methods are the same as beforeHa: Customers’ payment methods are not the same as before = 5%df = 3-1 = 2Reject Ho if the calculated > 5.991

fo fe

37 46 (23%) 1.7647 70 (35%) 7.56116 84 (42%) 12.19

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200 200 21.51 Reject HO: The customers’ payment methods are not the same as before.

29. The manager of a local Spar counted the number of customers using the store’s five checkout lanes during Friday and Saturday of a certain week. The results are as follows:

Checkout Lane1 2 3 4 5

Number of customers 160 200 300 120 100Lane 5 is closed much of the time because it is used during busy times only. The manager suspects, prior to taking the actual count, that lane 5 will be used half as often as lanes 1, 2 and 4. Checkout lane 3 is the express lane and is used by twice as many people than lanes 1, 2 and 4. Test the managers believe that certain lanes are used more than others using = 0.01.

Ho: The lanes are used the same number of timesHa: The lanes are not used the same number of times = 1%df = 5-1 = 4Reject Ho if the calculated > 13.277

fo fe

160 160 (2) 0200 160 (2) 10300 320 (4) 1.25120 160 (2) 10100 80 (1) 5880 880 26.25

Reject HO: Certain lanes are used more often than others.

30. Two companies have recently conducted aggressive advertising campaigns to maintain or increase their respective market shares for a particular product. Before the campaigns, the market share of company A was 45%, while company B had a share of 40%. Other competitors accounted for the remaining 15%. To determine whether these market shares changed after the campaigns, a market analyst determined the preferences of a random sample of 200 customers of this product. Of the sample, 100 indicated a preference for company A’s product, 85 preferred company B’s product and the remainder preferred another competitor’s product. Conduct a test to determine, at a 2.5% level of significance, whether the market shares have changed from the previous levels.

Ho: Market shares are still the sameHa: Market shares have changed = 2.5%df = 3-1 = 2

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Reject Ho if the calculated > 7.378 fo fe

100 90 (45%) 1.1185 80 (40%) 0.3115 30 (15%) 7.5200 200 8.92

Reject HO: Market shares have changed.

31. A computer store owner feels that 50% of her customers purchase word-processing programs, 25% purchase spread sheet programs and 25% purchase games. A sample of purchases shows the following distribution:

Word-processing Spread sheet Games

38 23 19

Test her assumption at a 10% level.

Ho: The store owner’s assumption is accurateHa: The store owner’s assumption is not accurate = 10%df = 3-1 = 2Reject Ho if the calculated > 4.605

fo fe

38 40 (50%) 0.123 20 (25%) 0.4519 20 (25%) 0.0580 80 0.60

Accept HO: The store owner’s assumption is accurate

32. Supermarket chains often carry products with their own brand labels and usually price them lower than the than the other brands. A supermarket conducted a taste test to determine whether there was a difference in taste among the four brands of ice cream it carries: own brand (A) and three other brands B, C and D. A sample of 200 people participated and they indicated their preference as shown in the table below. Test at a 5% level of significance if there is a difference in preference for the four brands.

A B C D39 57 55 49

Ho: No difference in taste between the four brandsHa: Is a difference in taste between the four brands = 5%df = 4-1 = 3

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Reject Ho if the calculated > 7.815 fo fe

39 50 2.4257 50 0.9855 50 0.5049 50 0.02200 200 3.92

Accept HO: No difference in taste between the four brands.

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Text book answers unit 11.doc

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