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Confidence Intervals and Sample Size

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1. Suppose a sample of n = 50 items is drawn from a population of manufactured products and the weight, X, of each item is recorded. Prior experience has shown that the weight has a probability distribution with m= 6 ounces and standard deviation= 2.5 ounces. Which of the following is true about the sampling distribution of the sample mean if a sample of size 15 is selected?
a) The mean of the sampling distribution is 6 ounces.
b) The standard deviation of the sampling distribution is 2.5 ounces.
c) The shape of the sample distribution is approximately normal.
d) All of the above are correct.

2. A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 99% reliability, how many students would need to be sampled?

a) n = 1,844
b) n = 1,784
c) n = 1,503
d) n = 1,435

3. An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What sample size would the economist need to use for a 95% confidence interval if the width of the interval should not be more than $100?
a) n = 1537
b) n = 385
c) n = 40
d) n = 20

I need detail explanation.

Thanks.

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