Question

A manufacturer produces both a deluxe and a standard model of an automatic sander designed for home use. Selling prices obtained from a sample of retail outlets follow.

	Model Price ($)				Model Price ($)
Retail Outlet	Deluxe	Standard		Retail Outlet	Deluxe	Standard
1	40	27		5	40	30
2	39	28		6	39	32
3	43	35		7	36	29
4	38	31

The manufacturer's suggested retail prices for the two models show a $10 price differential. Use a .05 level of significance and test that the mean difference between the prices of the two models is $10.

Develop the null and alternative hypotheses.
H 0 =  d Selectgreater than 10greater than or equal to 10equal to 10less than or equal to 10less than 10not equal to 10Item 1  
H a =  d Selectgreater than 10greater than or equal to 10equal to 10less than or equal to 10less than 10not equal to 10Item 2  

Calculate the value of the test statistic. If required enter negative values as negative numbers. (to 2 decimals).
  

The p-value is Selectless than .01between .10 and .05between .05 and .10between .10 and .20between .20 and .40greater than .40Item 4  

Can you conclude that the price differential is not equal to $10?
SelectYesNoItem 5  

What is the 95% confidence interval for the difference between the mean prices of the two models (to 2 decimals)? Use z-table.
( ,  )

Answer 1

The Null and the Alternative Hypothesis is obtained as:

i.e the mean difference between the prices of the two models is equal to $10.

against,

i.e the mean difference between the prices of the two models is not equal to $10.

Now, here,

and

Again, the sample variance is obtained as,

and,

Thus, the pooled mean is obtained as:

Thus, the test statistic is obtained as:

Now, the test statistic follows a t-distribution with degrees of freedom

Thus, the p-value is obtained as :

Thus, the p-value is greater than 0.4.

Conclusion: We observe that the p-value is greater than (at level of significance), thus we do not reject the null hypothesis and hence conclude that the mean difference between the prices of the two models is equal to $10.

Thus, the confidence interval for the difference between the mean, i.e , where , is obtained as:

Thus, the 95% confidence interval for the difference between the mean prices of the two models is obtained as

[The Z-Table is not used because the population variance or standard deviation is unknown, hence the test statistic under the hypotheses will follow t-distribution.]