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Suppose a set of data is symmetrically distributed about the mean x = 0. Assume that 𝑝(𝑥 < −1.2) = 0.32 and 𝑝(𝑥 > 2.1) = 0.29. Find the following probabilities. Show all necessary work on how you obtained your answer.
a. 𝑝(𝑥 ≥ −1.2)
b. 𝑝(𝑥 < −2.1)
c. 𝑝(−1.2 < 𝑥 < 2.1)
d. 𝑝(−2.1 < 𝑥 < −1.2)
2. Suppose that the number of cups of water that people drink during a given day approximately normal with a mean of 3.2 cups and a standard deviation of 0.6 cups. What is the probability that a person randomly selected will have between 2 cups and 3.8 cups of water per day?
3. The diameter of a rand of tennis balls is approximately normally distributed, with a mean of 2.63 inches and a standard deviation of 0.03 inches. If you select a random sample of nine tennis balls, what is the probability that the sample mean is between 2.65 and 2.59?
4. Suppose that the sampling error 𝜖 = 5 for a significance level of 0.28 and the sample mean is equal to 3. Construct a 72% confidence interval for the population mean. Clearly distinguish the bounds of the interval.
5. Suppose that the test scores for a test are normally distributed, and assume that a sample size of 17 was selected from the population. You calculated the mean to be 82 and the standard deviation of the sample to be 4.
a. Would you use a z- or a t-critical value? Why do you choose this?
b. What is the corresponding critical values for:
i. 𝑐 = 0.925
ii. 𝑐 = 0.95
iii. 𝑐 = 0.975
(Explain how you obtain these values in Excel)
c. Construct the 92.5%, 95% and 97.5% confidence intervals for the population mean test scores. Be organized in the calculation and presentation of each of your answers.
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