(a)State the null and alternative hypothesis that should be used to test each of the following claims.

i.The manufacturer of a new product claims that his product will increase output per machine by at least 38 units per hour. A line manager adopts the product on 18 of his machines, and finds that the average increase was only 35 units per hour with a standard deviation of 12.2 units. Is there evidence to doubt the manufacturer's claim at a significance level of .10?

(2 Marks)

ii.A fitness club claims that the average age of its members is 42. Assume that the ages are normally distributed. You believe that the average age is less than 42, therefore, you ask the ages of a few of the fitness club's members and get the following values: 40, 55, 50, 33, 38, 35, 30, and 45. Do you have any reason at the 10 per cent level of significance to think the fitness club's claim is incorrect?

(2 Marks)

(b)State the null and alternative hypotheses for each of the following examples. Determine if each is a case of a two-tailed, a left-tailed, or a right-tailed test.

i.An engineering research center claims that, through the use of a new computer control system, automobiles should achieve, on average, at least an additional 5 miles per gallon of gas. A random sample of 81 automobiles was used to evaluate this product. The sample mean increase in miles per gallon achieved was 4.5, and the sample standard deviation was 2.9 miles per gallon.

(3 Marks)

ii.A professional rock climber claims that he can climb a particular mountain within half an hour. From a random sample of 20 attempts, the rock climber averaged 33.3 minutes with a standard deviation of 3.9 minutes. Is there sufficient evidence to suggest that the climber's claim is incorrect at a significance level of 10 per cent?