Two suppliers manufacture a plastic gear used in a laser printer. the impact strength of these gears measured in foot-pounds is an important characteristic. a random sample of 10 gears from supplier 1 results in 1 = 290 and s1 = 12, while another random sample of 16 gears from the second supplier results in 2 = 321 and s2 = 22.(a) is there evidence to support the clain that supplier 2 provides gears with higher mean impact strength? use α = 0.05, and assume that both populations are normally distributed but the variances are not equal. what is the p-value for this test? (b) do the data support the claim that the mean impact strength of gears from supplier 2 is at least 25 foot - pounds higher than that of supplier 1? make the same assumptions as in part () construct a confidence interval estimate for the difference in mean impact strength, and explain how this interval could be used to answer the question posed regarding supplier-to-supplier differences.

a) Yes, there is. The p-value for this test is 0.0002

b) Yes, it does

c) [-32.642,-29.358]

Step-by-step explanation:

Since both populations are normally distributed but the sample sizes are less than 30, we will be using the Student's t Distribution for a comparison of the difference of means.

a)

We compute our t-statistic by the formula

where

= 290 is the mean of supplier 1

= 321 is the mean of supplier 2

= 10 is the sample size from supplier 1

= 16 is the sample size from supplier 2

is the pooled standard deviation

= 12 is the standard deviation of supplier 1

= 22 is the standard deviation of supplier 2

We have then

and out t-statistic is  

Our p-value is the area under the t Distribution with 24 degrees of freedom (sample1 -1 + sample2 -1) to the left of -4.073.

We can find this value by using either a table or a spreadsheet.

In Excel use

TDIST(4.073,24,1)

In OpenOffice Calc use

TDIST(4.073;24;1)

We get a p-value = 0.0002

b)

Since 321 is 31 pounds higher than 290 and our p-value is less than the significance level 0.05, the data do support the claim that the mean impact strength of gears from supplier 2 is at least 25 foot - pounds higher than that of supplier 1.

c)

A 95% confidence level interval correspondent to a significance level of 0.05 is given by

This means that there is a 95% probability that the mean impact strength of gears from supplier 2 is between 29.358 and 32,642 foot - pounds higher than that of supplier 1