Inference about the difference between two population means - Independent samples, population standard deviation unknown

Most consumers are not diamond experts, so many rely on an independent certification body to determine a diamond's value. Diamonds are assessed on four characteristics, referred to as the "four Cs": carat weight, color, clarity, and cut. Several certification bodies issue diamond grading reports, including the Gemological Institute of America (GIA), the Hoge Raad voor Diamant(HRD), and the International Gemological Institute (IGI).

If a certification body is recognized as strict in its grading, diamonds certified by that body may sell for higher prices than diamonds with the same grades that are certified by a different body. Compare the prices of diamonds graded by one certifier to the prices of diamonds graded by a different certifier to learn whether one population carries a price premium. The Diamonds data set in the following DataView tool contains data on the carat weight, color, clarity, certification body, and selling price of a random sample of 308 round-cut diamond stones. [Source: Singfat Chu, Journal of Statistics Education Data Archive.]

Define population 1 as diamonds certified by HRD and population 2 as diamonds certified by GIA. Similarly, define as the mean price of diamonds certified by HRD and as the mean price of diamonds certified by GIA.

The point estimate of - is _____ . (Hint:To use the DataView tool to see statistics for only one certification body, click on one of the Variable sliding panels, and select Certification Body in the dropdown menu. Exclude diamonds certified by the other two certifiers by unchecking the boxes next to those. Then click on another Variable sliding panel, and select Price in the dropdown menu. Find the values you are seeking in the Filtered columns. For example, the sample mean price for diamonds certified by HRD is $7,181.)

The population standard deviations and are unknown. Use the Distributions tool and the DataView tool to develop a 99% confidence interval estimate of the difference between the mean prices of diamonds certified by the two certifiers.

The 99% confidence interval estimate of the difference between the two population means is ___ to ____ . (Hint:The appropriate degrees of freedom is 174.)

Based on your interval estimate, a two-tailed hypothesis test of H: - = 0 conducted at the 0.01 level of significance would _____ H, because the value 0 is _____ in the 99% confidence interval estimate of - .

Maybe consumers don't perceive HRD to be stricter in its grading than GIA. It might be the case that prices of diamonds certified by the two certifiers are different because HRD certifies more large diamonds (which sell for higher prices than small diamonds) than GIA.Now redefine as the mean weight of diamonds certified by HRD and as the mean weight of diamonds certified by GIA. Using a significance level of = 0.01, conduct a hypothesis test to determine whether diamonds certified by HRD have a higher mean weight than diamonds certified by GIA. The appropriate degrees of freedom is 201.

You conduct ____ test with the null and alternative hypotheses formulated as:

a. H: - 0; Haa: - 0

b. H: - 0; Haa: - > 0

c. H: - 0; Haa: - = 0

d. H: - 0; Haa: - < 0

The value of the test statistic is _____ .

The p-value is ____ . Therefore, you _____ conclude that diamonds certified by HRD have a higher mean weight than diamonds certified by GIA.