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Probability and Statistical Analysis of Normal Distributions

An assignment covering probability, normal distributions, and statistical analysis.

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    Probability and Statistical Analysis of Normal Distributions 1. Find the following probability for the standard normal random variable z P( - 2.5 ≤ z ≤ 2.5) = 0.9876 2. Suppose x is a normal distributed random variable with μ = 15 and = 2. Find each of the following probabilities. a. P(x ≥ 18.5) = 0.0401 b. P(x ≤ 11) = 0.0228 c. P(16.56 ≤ x ≤ 19.58) = 0.2067 d. P(9.52 ≤ x ≤ 18.6) = 0.9610 3. The ages of a group of 50 women are approximately normally distributed with a mean of 50 years and a standard deviation of 5 years. One woman is randomly selected from the group, and her age is observed. a. Find the probability that her age will fall between 55 and 60 years. 0.1359 b. Find the probability that her age will fall between 49 and 53 years. 0.3050 c. Find the probability that her age will be less than 34 years. 0.0007 d. Find the probability that her age will exceed 41 years 0.9641 4.Financial analysts who make forecasts of stock prices are categorized as either “buy - side” analysts or “sell - side” analysts. The mean and standard deviation of the forecast errors for both types of analysts are shown is the table to the right. Assume that the distribution of forecast errors are approximately normally distributed. Buy - side analysts Sell - side analysts Mean .82 - .03 Standard deviation 1.94 .86 a. Find the probability that a buy - side analyst has a forecast error of + 2.02 or higher 0.2681 b. Find the probability that a sell - side analyst has a forecast error of + 2.0 2 or higher 0.0000 4. The mean gas mileage for a hybrid car is 56 miles per gallon. Suppose that the gasoline mileage is approximately normally distributed with a standard deviation of 3.2 miles per gallon.

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