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 µi of 0.99 carats diamonds is lower than the true average price µ2 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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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!