A vocabulary test for six-year-old children is standardized on a large nationwide population to have an average score of 50 out of 100, with a standard deviation of 10.

School authorities in Maine choose a statewide simple random sample of 400

six-year-olds. These children average 51.3 on the test.Is it safe to infer that, if the test had been administered to all six-year-olds in Maine, they would have averaged above 50? (Assume Maine's scores have an equal standard deviation as the nation, = 10). Use = 0.01.

a.Formulate the null and alternative hypotheses and state assumptions.

b.Compute test statistic.

c.State the decision rule in terms of the test statistic for testing hypotheses and your conclusion in terms of the problem (not merely "reject" or "accept" Ho).

d.Find the P-value.

e.Find the 95% CI for (the population mean) if the test had been administered to all six-year-olds in Maine.

f. How many observations would be required to ensure that a 99% confidence interval for would have the same length as Part e's Interval?[Again, assume = 10.]

Please assist. I understand its more than one (total of 5 questions) to assist me with. Your help is really appreciated. Please show step by step answers. Thank you.