Refer to the below questions and kindly assist with showing the calculation and with a clear explanation.

Question 1

A post office processes an average of 35 letters per day. Find the probability that the office will process exactly 30 letters in a day.

Question 2

The nap time for kids in kindergarten follows normal distribution with a mean of 25 minutes and a standard deviation of 5 minutes. Find the probability that the nap time for kids is

a) greater than or equal to 30 minutes. (4 marks)

b) between 20 to 30 minutes. (4 marks)

Question 3

Consider a null hypothesis of Ho: u=8. A random sample of 150 observations is taken from a population derives a variance of 289 and mean of 10. By using 0.05 significant level, conduct a test on the hypothesis and conclude your result.

Question 4

The average amount of rice put into plastic packages when automatic filler is set is 5,000 grams. A sample of 100 packages was chosen at random yields the following result.

Mean = 4,930 grams; standard deviation = 3,000 grams

Suppose that the producer wants to test the hypothesis that the average rice content is different from 5,000 grams.

a) State the null and alternate hypothesis (10 marks)

b) Based on the observed value of the test statistics, with alpha = 0.05, what can be concluded? (10 marks)

HINTS: alpha = 0.05, the critical value is z = 1.65 for one-tail; or z = 1.96 for two-tails, and the formulae Z = x-

standard deviation/ square root of n