Please show your work.

1. The average federal income tax paid by a family of four earning $38,000 per year is to be estimated. For this purpose, taxes paid by 50 such families were sampled from federal records. The mean and standard deviation were found to be $4,288 and $268, respectively. Find the 90% confidence interval.

2. A professor of Statistics feels that the performance level of students has gone down recently. In the past, final exam scores used to have a mean of of 75. This semester, in a class of 32 students the average score is 71 with a standard deviation of 8. Does this really justify the professor's concern?

(a) State the null hypothesis and alternative hypothesis

(c) If the type-I error = 0.05, find the critical value(s) and shade the rejection region(s)

(b) Find the test statistic

(d) Base on the type-I error and rejection region(s), given above, what is your conclusion?

(e) Find the p-value of test statistic

4. A certain type of house plant has leaves averaging in length of 7 cm. Lisa wanted to try a new fertilizer from Miracle Grow which claims to improve the size of plant leaves. After two months of using the new fertilizer, she measured this sample of values:

6.2, 6.5, 7.2, 7.5, 7.7, 7.1, 7.8, 7.9, 8.2, 8.1

(a) State the null hypothesis and alternative hypothesis

(b) Find the test statistic

c) If the type-I error = 0.05, find the critical value(s) and shade the rejection region(s)

(d) Base on the type-I error and rejection region(s), given above, what is your conclusion?