Prices of diamonds are very sensitive to their weight (measured in carat). For example, the price of a 1 carat diamond tends to be much higher than the price of a 0.99 carats diamond. To study this phenomenon in more detail, we consider two random samples of diamonds, 0.99 carats and 1 carat, each sample of size 23. The average price of 0.99 carats diamonds in the sample is $4451, while the average price of 1 carat diamonds in the sample is $5681. From long-term studies, we know that both the prices of 0.99 carats diamonds and the prices of 1 carat diamonds are normally distributed with standard deviations ơ1 = $1332 (0.99 carats) and o2 = $1613 (1 carat), respectively. (a) Carry out a hypothesis test at significance level 0.01 to evaluate whether the true average price µ1 of 0.99 carats diamonds is lower than the true average price µz of 1 carat diamonds. Specify Ho and Ha, the test statistic, and calculate the p-value or use p-value considerations to conclude. (b) Compute the type II error probability for the test in (a) if the true average price of 1 carat diamonds is $5500 and the true average price of 0.99 carats diamonds is $4600. Include a detailed derivation in your answer. (c) How large do the sample sizes (kept equal for 0.99 carats and 1 carat diamonds) have to be such that, in the situation described in (b), the type II error probability is at most 5%? Include a detailed derivation in your answer.

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.