Question

   Once again, as has happened in the past, you are very much in doubt concerning the validity of the known population standard deviations, this time for each camera body, in the yearly sales of the two brands of camera bodies. Therefore, you wish to conduct your study with the knowledge that the population standard deviations are unknown. You collect random samples of the yearly sales of the two camera bodies at populations of stores. The data that has been collected is shown in appendix one below. At both the 10% and 5% levels of significance, are there any differences in the mean sales of the two camera bodies at the two populations of stores? Again, if the software makes it possible, find both 90% and 95% confidence intervals for the difference in the mean sales of the camera bodies between the two populations of stores. Explain the meanings of these intervals. Then, if possible, based upon the procedures you have chosen to address the problem, use the intervals to supplement and test whether there is a difference in the mean sales of the two camera bodies between the two populations of stores.

Appendix One: (Sales of Camera Bodies)

Nikon D5:

131 145 150 156 176 154 138 122 130 235 165 168 221 229 154 155 154 160 154 144 240 143 232 238 130

Canon Model:

      138 140 237 147 170 155 232 228 135 130 161 160 220 229 155 158 150 250 248 246 139 233 133 230 126     

Appendix Two: (Includes the Purchase of a Lens? Y = yes, N = no)

Nikon D5:

      Y N N N Y Y N Y N Y Y N N N N Y Y Y N Y Y N N N N N N Y N N N Y N N N N

      

Canon Model:

      N N Y Y Y N N Y N Y N N Y Y Y Y N N N N Y N Y N Y N N N Y Y Y Y Y N Y Y

Answer 1

We are given the following data and are asked to test the hypothesis that there is a difference in the mean sales of both the camera bodies and also find the confidence interval at 90% and 95% confidence.

Nikon D5:

131 145 150 156 176 154 138 122 130 235 165 168 221 229 154 155 154 160 154 144 240 143 232 238 130

Canon Model:

138 140 237 147 170 155 232 228 135 130 161 160 220 229 155 158 150 250 248 246 139 233 133 230 126.

We now have to calculate mean and standard deviation for the two camera bodies.

Let Nikon D5 be denoted by x and Canon Model be denoted by y

xi	xi2	yi	yi2
131	17161	138	19044
145	21025	140	19600
150	22500	237	56169
156	24336	147	21609
176	30976	170	28900
154	23716	155	24025
138	19044	232	53824
122	14884	228	51984
130	16900	135	18225
235	55225	130	16900
165	27225	161	25921
168	28224	160	25600
221	48841	220	48400
229	52441	229	52441
154	23716	155	24025
155	24025	158	24964
154	23716	150	22500
160	25600	250	62500
154	23716	248	61504
144	20736	246	60516
240	57600	139	19321
143	20449	233	54289
232	53824	133	17689
238	56644	230	52900
130	16900	126	15876

Since population standard deviations are not known, we use pooled standard deviation.

Null hypothesis

i.e., there is no difference in the mean sales of both the camera parts

Alternate hypothesis

i.e., there is a difference in the mean sales of both the camera parts

Test statistic under H₀:

Inference

at 5% significance level, 48 degrees of freedom and for two-tailed test from t-table

=> We fail to reject the null hypothesis.

Hence we conclude that there is no difference in the mean sales of the two camera bodies.

at 10% significance level, 48 degrees of freedom and for two-tailed test from t-table

=> We fail to reject the null hypothesis.

Hence we conclude that there is no difference in the mean sales of the two camera bodies.

The formula for calculating 90% confidence interval for the difference in the mean sales of the two camera bodies is,

=>

=>

Hence the 90% confidence interval is [-4.04,30.12]

This means that we can be 90% confident that the true mean difference lies between -4.04 and 30.12.

For 95% confidence only the critical value will change, the formula remains the same.

=>

Hence the 95% confidence interval is [-7.40, 33.48]

This means that we can be 95% confident that the true mean difference lies between -7.40 and 33.48.

Note:- I made a mistake here, I took the smaller mean as the first mean i.e., x-bar should have been mean of Canon Model and y-bar should have been mean of Nikon D5, this does not change the result of the hypothesis test but it could have affected our confidence intervals. So I interchanged the means so that we could get the correct intervals but the rest remains the same.