Please explain the theory involved first, then step by step, detailed explanations ,with good presentations, answer and explanations to all parts please, please leave the question if you cant , as it would hurt your profile, thank you!

A manufacturer of detergent claims that the contents of boxes sold weigh on average 16 ounces. The distribution of weight is known to be normal with a population standard deviation of 0.4 ounces. A random sample of 25 boxes is collected. 31. Using a significance level a = 0.10, identify the range of possible sample means where you would fail to reject the null hypothesis that the population mean is 16 ounces against a two-sided alternative. b. Identify the range of sample means such that the 90% confidence intervals based on those means will contain 16 ounces. A pharmaceutical manufacturer is aware that for safety issues, the impurity concentration in their pills should not exceed 3% on average. Suppose that the population standard deviation is known, and it is a = 0.4%. A random sample of 64 pills was checked and the sample mean impurity concentration was found to be 3.07%. a. Using a significance level a: = 0.05, test the hypothesis that the population mean impurity concentration is 3% against the alternative that it is more than 3%. Identify the p-value for this test. b. Suppose that the alternative hypothesis had been twosided rather than one-sided. Without doing any calculations, would the p-value be higher than, lower than, or the same as that found in part (a). Sketch a graph to illustrate your reasoning. c. Based on your answer in part (b), is it easier to reject the null hypothesis with a one-sided or a two-sided hypothesis test