Statistics - Normal Probability Projects

For a random variable that is normally distributed, with µ = 80 and ơ = 10, determine the probability

For a random variable that is normally distributed, with µ = 80 and ơ = 10, determine the probability that a simple random sample of 25 items will have a mean that is:

a) greater than 78.

b) between 79 and 85.

c) less than 85.
It has been estimated that 17.0% if mutual fund shareholders are retired persons. Assuming the population proportion to be π = 0.17, and that a simple random sample of 400 shareholders has been selected:

a) what is the expected value of p = the proportion of those in the sample who are retired?

b) what is the standard error of the sampling distribution of the proportion, p?

c) what is the probability that at least 20% of those in the sample will be retired?

d) what is the probability that between 15% and 25% of those in the sample will be retired?
A machine that stuffs a cheese-filled snack product can be adjusted for the amount of cheese injected into each unit. A simple random sample of 30 units is selected, and the average amount of cheese injected is found to be x = 3.5 grams. If the process standard deviation is known to be ơ = 0.25 grams, construct the 95% confidence interval for µ = the average amount of cheese being injected by the machine.
Given the following observations in a simple random sample from a population that is approximately normally distributed, construct and interpret the 90% and 95% confidence intervals for the mean:

67 79 71 98 74 70 59 102 92 96
From past experience, a package-filling machine has been found to have a process standard deviation of 0.65 ounces of product weight. A simple random sample is to be selected from the machine's output for the purpose of determining the average weight of product being packed by the machine. For 95% confidence that the sample mean will not differ from the actual population mean by more than 0.1 ounces, what sample size is required?