Please show the work and formula used and answers in numerical values. Thanks!

1. The sugar content of the syrup in canned peaches is normally distributed. A consumer wants to estimate the standard deviation of sugar content in these cans. A random sample of 10 cans yields a standard deviation of 4.8 milligrams. (a) Give a point estimate for the population standard deviation, 0. (b) Find a 95% condence interval for the population standard deviation. Give a practical interpretation for the answer. 2. A manufacturer of electronic calculators is interested in estimating the proportion of defective units produced. A random sample of 800 calculators contains 10 defective. (a) Give a point estimate for the proportion of defective units. (b) Compute a 99% (two-sided) confidence interval for the proportion of defective units. (c) Suppose we want a 95% confidence interval with error not exceeding 0.005. How big a sample do you need to achieve this accuracy?3. Suppose you plan to carry out a test of a null hypotheses of H0 against the alternative hy- pothesis H1 based on a random sample and make a decision. (a) If you decide to reject H0 (in favor of H1), what type of error you may possibly make, Type I error, Type II error, or neither? (b) If you decide not to reject H0, What type of error you may possibly make, Type I error, Type II error, or neither? 4. All the Test of the questions below are based on the following context. A textile ber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. The company wishes to test the hypothesis H0 : a = 12 against the alternative H1 : ,u > 12 using a random sample of four specimens. (a) Find the probability of type I error probability when the critical (i.e., rejection region) region is: E > 12.5 kilograms. (b) Give the definition of Type II error, and use it to find the probability of Type II error when u = 14. (c) For the questions bellow, assume that the sample mean, x, is 12.7.(i) Carry out the test of the hypothesis Ho : / = 12 against the alternative H1 : u > 12 at 5% significance level, showing all steps: (i) Sate null and alternative hypotheses, (ii) Give formula for test statistic, and calculate its value, (iii) State the decision rule, (iv) Make the conclusion and state it in the context of the problem. (ii) What is the p-value