Hypothesis Testing Questions 
Practice Questions
1. The director of manufacturing at a clothing factory needs to determine whether a new machine is producing a particular type of cloth according to the manufacturer's specifications, which indicate that the cloth should have a mean breaking strength of 32 kilograms and a standard deviation of 2.72 kilograms. A sample of 49 pieces of cloth reveals a sample mean breaking strength of 31.3 kg. ( solution )
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2. According to a newspaper article, "More than threequarters of Canadians believed the quality of the environment declined over the last decade…" This statement was based on a poll in which 1161 of the 1504 Canadians who responded agreed. Is there sufficient evidence to conclude that over 75% of the population agreed with the statement? Test at the 5% level of significance. ( solution )
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3. A machine that fills bottles of salad dressing at a food plant is working properly when it fills each bottle with 250 millilitres of liquid dressing mix. The manufacturer must be sure that the bottles contain on average no less than 250 millilitres. The standard deviation of the process is 4.5 millilitres. A sample of 50 bottles is selected periodically, and the filling line is stopped if there is evidence that the mean fill has dropped to less than 250 millilitres. Suppose the average fill is 249.5 millilitres from a sample of 50 bottles. ( solution )
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4. A trucking firm suspects the claim that the average lifetime of certain tires is at least 28,000 miles. To check the claim, the firm tests 40 such tires on its trucks and gets a mean lifetime of 27,563 miles with a standard deviation of 1,348 miles. Perform the test at the 0.01 level of significance. ( solution )
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5. A law student wants to test the claim that convicted embezzlers spend an average of 12.3 months in jail. He tests a random sample of 35 such cases and gets an average of 11.5 months with s = 3.8 months. Test at the 0.05 level of significance. ( solution )
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6. A diesel engine manufacturer claims that his engines have a mean thermal efficiency of at least 32.3%. Tests performed on 40 of his engines yielded a mean thermal efficiency of 31.8% with a standard deviation of 2.2%. Test his claim at the 0.05 level of significance. Find the tail probability or pvalue associated with these test results. ( solution )
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7. A food study reports that 1 pound of goodfish yields an average of 2.41 ounces of FPC (fishprotein concentrate) used to enrich food. Is this average supported by a study
of 30 samples of goodfish which show an average of 2.44 ounces of FPC/lb. and a standard deviation of 0.07 ounces if we use the: ( solution )
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8. An automobile rustproofing company claims that their method protects cars
for an average of 55 months. This hypothesis is tested against the alternative
that the protection lasts for more than 55 months. A random sample of 200 cars
produces an average protection time of 56 months with a standard deviation of 15 months.
Test the claim at the 5% level of significance and draw a conclusion. ( solution )
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9. A deodorant manufacturer claims that the mean drying time of their product is at most
15 minutes. A sample of 16 cans yielded a mean drying time of 18 min. with s = 6 minutes.
( solution )
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10. The yield of alfalfa, in tons per acre, from 6 test plots of organic soil is : ( solution )
1.4  1.8  1.1  1.9  2.2  1.2 
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11. The owner of a gasoline station wants to study gasoline purchasing habits by motorists at his station. A random sample of 60 motorists in a certain week gave these results:
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12. A television critic claims that 80% of all viewers find the noise level of a certain commercial objectionable. If a random sample of 320 viewers includes 245 who find the noise level objectionable, do these results support his claim at the 5% level of significance? ( solution )
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Solutions
b) Solution: Since we are given a standard deviation, we use the ztest:
c) Solution:
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Hypothesis Test on Mean  
Value of interest: µ , mean fill of bottles  
Hypotheses: Ho:  Ha: l < 250 
Level of significance: = 5 %  Criteria: if z < – 1.645, reject Ho 
Test Statistic:

Decision: Since – 0.756 is not less than – 1.645, we can't reject Ho 
Conclusion: the test indicates there is not enough evidence to prove the bottles are being filled with less than 250 mL of salad dressing. 
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Lower Tail Test on a Mean  
Ho: l m 28,000 miles  Ha: l < 28,000 miles 
Level of significance: = 1 %  Criteria: if z < – 2.33, reject Ho 
Statistic:  
Decision: we cannot reject the null hypothesis Ho. Conclusion: we conclude that the average lifetime of certain tires is at least 28,000 miles. 
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2tail test on mean length of jail sentence for embezzlers.  
Ho:µ = 12.3 months  Ha: months 
Level of significance: = 5 %  Criteria: if  z  > 1.96, reject Ho 
Statistic:  
Decision: we cannot reject the null hypothesis Ho. Conclusion: the claim that convicted embezzlers spend an average of 12.3 months in jail cannot be refuted by the results of this test. 
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Lower Tail Test on a Mean  
Ho: l m 32.3 % efficiency rating  Ha: l < 32.3 % efficiency rating 
Level of significance: = 5 %  Criteria: if z < – 1.645, reject Ho 
Statistic:  
Decision: we cannot reject the null hypothesis Ho Conclusion: his engines have a mean thermal efficiency of at least 32.3% the pvalue = the probability of being in the tail created by the statistic z = – 1.44 
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a) 0.05 level of significance; .......... b) 0.01 level of significance?
2Tail Test on a Mean  
Ho:µ = 2.41 ounces  Ha: ounces 
Level of significance: a) = 5 %,

Criteria: a) if  z  > 1.96, reject Ho

Statistic :  
Decision: a) since 2.35 > 1.96, we must reject the null hypothesis Ho.

c) Solution:
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value of interest: mean protection time from rustproofing.  
Ho: l = 55  Ha: l > 55 
Level of significance: = 5 %  Criteria: if z > 1.645, reject Ho 
Statistic:  
Decision: we cannot reject the null hypothesis Ho. Conclusion: it seems that the rustproofing protection does last for an average of 55 months. 
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hypothesis test on mean drying time (small sample)  
Ho: l [ 15  Ha: l > 15 
Level of significance: = 5 %, df = 15  Criteria: if t > 1.753, reject Ho 
Statistic:  
Decision: we must reject the null hypothesis Ho Conclusion: the test shows the maximum drying time is more than 15 minutes. 
b) Solution:
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hypothesis test on mean yield of alfalfa (small sample)  
s by formula = 0.434  
Ho: µ = 1.5  Ha: 
Level of significance: = 5 %, df = 5  Criteria: if  t  > 2.571, reject Ho 
Statistic:  
Decision: we cannot reject the null hypothesis Ho Conclusion: the test shows that the average yield for this kind of alfalfa in this type of organic soil is 1.5 tons per acre. 
hypothesis test on mean volume of gasoline purchased  
s = 6.7 litres  
Ho: µ = 40 litres  Ha: 
Level of significance: = 5 %,  Criteria: if  z  > 1.96, reject Ho 
Statistic:  
Decision: since 3.24 > 1.96, we must reject the null hypothesis Ho Conclusion: the test shows that the mean gasoline purchase is different from 40 litres. 
Ho: p = 0.20  Ha: p < 0.20 (1tail test) 
Level of significance:  Criteria: if z < – 1.645, reject Ho 
Statistic:  
Decision: – 0.329 > – 1.645, so we cannot reject the null hypothesis Conclusion: we can't assume that fewer than 20% of the motorists buy premiumgrade gas. 
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Ho: p = 0.80  Ha: (2tail test) 
Level of significance:  Criteria: if  z  > 1.96, reject Ho 
Statistic:  
Decision: – 1.55 is > – 1.96, so we cannot reject the null hypothesis Conclusion: the critic's claim is probably true and 80% of the viewers find the noise in the ad hard to take. 
(all content of the MathRoom Lessons © Tammy the Tutor; 2004  ).