MS5312 Business Statistics-Inferences about Two Population Means

MS5312 Business Statistics-Inferences about Two Population Means

Exercises

Question 1
Independent random samples were selected from two populations, 1 and 2. The sample sizes, sample means and population variances were as shown in the following table. Find a 90% confidence interval for the difference in the population means, and interpret your result.

Population 1 Population 2
Sample size 64 64
Sample mean 2.9 5.1
Population variance 0.83 1.67

Question 2
How much combustion efficiency should a homeowner expect from an oil furnace? A report states that 80% or above is excellent, 75% to 79% is good, 70% to 74% is fair and below 70% is poor. A home-heating contractor who sells two makes of oil heaters (call them A and B) decided to compare their mean efficiencies.

An analysis was made of the efficiencies for 8 heaters of type A and 6 of type B. The efficiency ratings, in percentages, for the 14 heaters are shown in the table. Assume populations are normal with equal variances.

Type A 72 78 73 69 75 74 69 75
Type B 78 76 81 74 82 75

(a) Find a 90% confidence interval for (A  B), and interpret the result.
(b) Based on the result of part (a), would you conclude that there is a difference in the mean efficiency ratings for the two types of heaters? Explain.

Question 3
Independent random samples of n1 = 80 and n2 = 80 were selected from populations 1 and 2, respectively. The population parameters and the sample means and variances are shown in the accompanying table.

Parameters and statistics Population 1 Population 2
Population mean 1 2
Population variance

Sample size 80 80
Sample mean 11.6 9.7
Sample variance 27.9 38.4

(a) If your research objective is to show that 1 is larger than 2, state the null and alternative hypotheses that you would choose for a statistical test.
(b) Conduct the test for  = 0.1 and draw your conclusions. Do the data present sufficient evidence to indicate that 1 is larger than 2?
(c) Give the observed level of significance for the test, i.e. the p-value.


Question 4
In an effort to determine whether or not a new type of fertilizer is more effective than the type currently in use, researchers took 12 two-acre plots of land scattered throughout the county.
Each plot was divided into two equal-sized subplots, one of which was treated with the current fertilizer and the other of which was treated with the new fertilizer. Wheat was planted and the crop yields were measured. These data are listed below.

Plot: 1 2 3 4 5 6 7 8 9 10 11 12
Current fertilizer: 56 45 68 72 61 69 57 55 60 72 75 66
New fertilizer: 60 49 66 73 59 77 61 60 58 75 72 71

(a) Can we conclude at the 5% significance level that the new fertilizer is more effective then the current one?
(b) Estimate with 95% confidence the difference in mean crop yields between the two fertilizers.
(c) What is the required condition(s) for the validity of the results obtained in parts (a) and (b)?

Question 5
On 20 randomly selected cars, one of each type of tire is installed on the rear wheels and, as above, the cars are driven until the tires wear out. The number of miles until wear-out occurred is shown below. Can we conclude from these data that the new tire is superior?


Distance (in thousands of miles) until wear-out

New-design tire
57 64 102 62 81 87 61 62 74 62 100 90 83 84 86 62 67 40 71 77

Existing-design tire
48 50 89 56 78 75 50 49 70 66 98 86 78 90 98 58 58 41 61 82




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