Mountain States Surgery Center in Denver, Colorado, performs many knee replacement surgery procedures each year. Recently, research physicians at Mountain States have developed a surgery process they believe will reduce the average patient recovery time. The hospital board will not recommend the new procedure unless there is substantial evidence to suggest that it is better than the existing procedure. Records indicate that the current mean recovery rate for the standard procedure is 142 days, with a standard deviation of 15 days. To test whether the new procedure actually results in a lower mean recovery time, the procedure was performed on a random sample of 36 patients and found a sample mean of 140.2 days. Conduct a hypothesis test with the probability of making a Type I error is 5%.

1. State the null and alternate hypotheses
2. State the level of significance (alpha)
3. Compute the value of the test statistic
4. Determine the critical value and the decision rule
5. State your conclusion
6. Findthepvalue

AND

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.

1. 1. What percentage of the boxes will be underweight?
   2. The companys 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?
   3. The company president vetoes that plan, saying the company should give away less free cereal, not more. Her goal is to set the machine no higher than 16.2 oz and still have only 4% underweight boxes. The only way to accomplish this is to reduce the standard deviation. What standard deviation must the company achieve AND what does that mean about the machines?

I need to show all my work!