please show ALL work for statistics, along with formulas shown. thank you!

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You should work individually. If signs of collaboration with other students is detected, your exam will be void and a grade of 0 will be assigned to this part of the final. •

*You must show your work in all sections of the questions. Stating the final answer without showing your work will not get any credit.*

A cereal manufacturer has a machine that fills the boxes. Boxes are labeled “16 oz”, so the company wants to have that much cereal in each box. But since no packaging process is perfect, there will be minor variations. If the machine is set at exactly 16 oz and the Normal distribution applies (or at least the distribution is roughly symmetric), then about half of the boxes will be underweight, making consumers unhappy and exposing the company to bad publicity and possible lawsuits. To prevent underweight boxes, the manufacturer has to set the mean a little higher than 16.0 oz. Based on their experience with the packaging machine, the company believes that the amount of cereal in the boxes fits a Normal distribution with a standard deviation of 0.2 oz. The manufacturer decides to set the machine to put an average of 16.3 oz in each box. Use this information to answer the following questions.

a) What percentage of the boxes will be underweight?

b) The company’s lawyers say that the percentage you found in part (1) is too high. They insist that no more than 4% of the boxes can be underweight. So, the company needs to set the machine to put a little more cereal in each box. What mean should they set the machines at to achieve the 4% being underweight?

It is hoped that a newly developed pain reliever will more quickly produce perceptible reduction in pain to patients after minor surgeries than a standard pain reliever. The standard pain reliever is known to bring relief in an average of 3.5 minutes with standard deviation 2.1 minutes. To test whether the new pain reliever works more quickly than the standard one, 50 patients with minor surgeries were given the new pain reliever and their times to relief were recorded. The experiment yielded sample mean of 3.1 minutes and sample standard deviation of 1.5 minutes. Is there sufficient evidence in the sample to indicate, at the 5% level of significance, that the newly developed pain reliever does deliver perceptible relief more quickly?

a) Identify the null and alternative hypotheses

b) Specify the Type I and Type II errors under the specified hypotheses from a.

c) Specify the decision rule (Be specific in choosing your critical value)

d) Calculate test statistics from the sample data

e) Form Decision and state your conclusion.

A random sample of Math SAT Scores of 20 applicants to the Isenberg School of Management are reported with a mean of 510 and a variance of 930. Assume that the Math SAT scores for Isenberg applicants are normally distributed.

a) Construct a 90% confidence interval for the sample.

c) If we changed it to a 98% confidence level, what would the margin of error be? (2 decimal places)

d) ***This part is not related to part C. Now assume that we know that the population standard deviation is 50. If we wanted a 98% confidence level and the margin of error to be 10, how big of a sample of Isenberg applicants would we need to look at?

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