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Normal Probability & Sampling Distribution of Sample mean

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Given a normal distribution with a population mean of 100, a standard deviation of 10 and a sample size of n = 25, what is the probability that X is

a. less than 95?
b. between 95 and 97.5?
c. above 102.2?
d. There is a 65% chance that X is above what value?

Time spent using e-mail per session is normally distributed, with a population mean of 8 minutes and standard error of the mean is 2 minutes. If you select a random sample of 25 sessions,

a. what is the probability that the sample mean is between 7.8 and 8.2 minutes?
b. what is the probability that the sample mean is between 7.5 and 8 minutes?
c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 7.8 and 8.2 minutes?
d. Explain the difference in the results of (a) and (c).

You plan to conduct a marketing experiment in which students are to taste one of two different brands of soft drink. Their task is to correctly identify the brand tasted. You select a random sample of 200 students and assume that the students have no ability to distinguish between the two brands. (Hint: If an individual has no ability to distinguish between the two soft drinks, then the two brands are equally likely to be selected.)

a. what is the probability that the sample will have between 50% and 60% of the identifications correct?
b. the probability is 90% that the sample percentage is contained within what symmetrical limits of the population percentage?
c. what is the probability that the sample percentage of correct identifications is greater than 65%?
d. which is more likely to occur more than 60% correct identifications in the sample of 200 or more than 55% correct identifications in a sample of 1,000? Explain.

The fill amount of bottles of a soft drink is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liter. If you select a random sample of 25 bottles, what is the probability that the sample mean will be

a. between 1.99 and 2.0 liters?
b. below 1.98 liters?
c. greater than 2.01 liters?
d. the probability is 99% that the sample mean amount of soft drink will be at least how much?