A cereal company claims that their "honey and oat" product have a mean weight of 5 ounces following a normal distribution. To test this hypothesis, you select a random sample of 16 bags and find that the mean and standard deviation of this sample are 4.7 ounces and 0.5 ounces, respectively.

Use a significance level 0.01 to test the above hypothesis. What is your conclusion? state the null and alternative hypotheses?

Illustrate the critical region for the above test.

Find a 90% confidence interval for the population variance.

Assume in the main problem description, the population standard deviation was known to be 0.7. Using this new information, calculate the p-value of the test based on the same sample information and hypothesis. Based on the calculated p-value and the significance level of 0.1, what conclusion do you make?