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4. Inference about the difference between two means (unequal variances) Aa Aa 8 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, colour, 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 recognised 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, colour, 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.] Data Set Diamonds Sample Variables = 5 Observations = 308 Pricing the C's of Diamond Stones Singfat Chu, Journal of Statistics Education Data Archive Observations Variables Variable V Type V Form V Values V Missing V Observations Weight Quantitative Numeric 308 C Color Qualitative Nonnumeric 308 Clarity Qualitative Nonnumeric 308 Certification Body Qualitative Nonnumeric 308 Variable Price Quantitative Numeric 308 Variable Variable Variable Correlation Correlation Define population 1 as diamonds certified by GIA and population 2 as diamonds certified by IGI. Similarly, define 1 as the mean price of diamonds certified by GIA and p2 as the mean price of diamonds certified by IGI. The point estimate of H1 - H2 is .( Hint: Use the DataView tool to obtain the statistics that form the point estimate.)t Distribution Degrees of Freedom = 193 O O -5 3 The population standard deviations of and oz are unknown. Assume that the population variances are unequal. Use the Distributions tool and the DataView tool to develop a 90% confidence interval estimator of the difference between the mean prices of diamonds certified by the two certifiers. The 90% confidence interval estimate of the difference between the two population means is LCL = to UCL = ( Hint: The appropriate degrees of freedom is 214.) Based on your interval estimate, a two-tail hypothesis test of Ho: Hi - 2 = 0 conducted at the .10 level of significance would Ho, because the value 0 is in the 90% confidence interval estimate of wi - Hz. Maybe consumers don't perceive GIA to be stricter in its grading than IGI. It might be the case that prices of diamonds certified by the two certifiers are different because GIA certifies more large diamonds (which sell for higher prices than small diamonds) than IGI. Now redefine pi as the mean weight of diamonds certified by GIA and uz as the mean weight of diamonds certified by IGI. Using a significance level of a = .10, conduct a hypothesis test to determine whether diamonds certified by GIA have a higher mean weight than diamonds certified by IGI. The appropriate degrees of freedom is 174. Again, of and oz are unknown, and you can assume that they are unequal. You conduct test with the null and alternative hypotheses formulated as: O Ho : H1 - 12 = 0; H1: H1 - Hz > 0 Ho: H1 - 12 = 0; H1: H1 - Hz