Question

You have now been asked to study the yearly mean sales of cameras of two competing models at stores throughout the United States. You will also study the proportions of cameras sold that include certain lenses at a large store that sells both lenses. The specific questions you will be asked to answer are stated below. In addition, appropriate sample data for the studies you will be accomplishing are given below. Answer the following questions concerning the situations posed.

1. You first wish to know whether there is a difference in the yearly mean sales of two different camera bodies, the Nikon D5 and a competing Canon model. The population standard deviation in the sales of the Nikon model is still thought to equal 50 units. The population standard deviation in the sales of the competing Canon model is thought to equal 40 units. Random samples of 35 stores that sell the Nikon D5 and of 40 stores that sell the competing Canon model are selected. The sample mean number of Nikon D5 bodies that is sold is found to be 177.5. The sample mean number of Canon bodies that is sold is found to equal 154.4. At each of the 10% and 5% levels of significance, is the mean number of Nikon D5 bodies different from the mean number of Canon bodies sold for these two populations of stores? If, at either level of significance, you observe that there is a difference in the yearly mean sales between the populations of stores, is that difference different from 5 camera bodies? If the software makes it possible, find also both 90% and 95% confidence intervals for the difference in the yearly mean sales of the two camera bodies. Explain their meanings. Then, use these intervals to answer all previous questions posed in the problem that were addressed using hypothesis testing techniques. Use the 90% confidence interval to test at the 10% level of significance and the 95% confidence interval to test at the 5% level of significance.

Answer 1

Estimating mean population sales of Nikon -

₁ =

Here = 177.5, = 50, n = 35

Z at 90% = 1.645, at 95% = 1.96

₁ at 90% = = [163.6, 191.4]

₁ at 95% = = [160.9, 194.1]

Similarly, Estimating mean population sales of Canon -

2 =

Here = 154.4, = 40, n = 40

Z at 90% = 1.645, at 95% = 1.96

2at 90% = = [144.0, 164.8]

2at 95% = = [142, 166.8]

Since the sample mean for Canon (154.4) does not lie in population mean interval at both 90% and 95% and vice-versa - We can say that population means are different.

To find a confidence interval for the difference in yearly sales of two cameras, we need to create a new distribution

which is given by Nikon - Canon

E(X₃) = E(X₁) - E(X₂) = 177.5 - 154.4 = 23.1

₃ = = 64.03

We get the Confidence intervals at 90% and 95% as follows

At 90% - [5.3, 40.9]

At 95% - [1.9, 44.3]

This signifies that the difference between the two population mean would be somewhere between 5.3-40.9 at 90% confidence and 1.9-44.3 at 95% confidence