5.9 Assignment 5

Purposes

This assignment has two parts. The first part assesses your knowledge of the properties of normal density curves, calculating probabilities related to normal distributions, and finding the quantiles of normal distributions. The second part assesses your skills in using R commander to find probabilities and quantiles of normal distributions.

Resources

M05__MonthlyFees _labQ4.xlsx

Instructions

Part A

Complete the following:

1. The standard normal distribution has mean _____ and standard deviation _______. (2 marks)
2. The area under the density curve between 30 and 40 is 0.832. What percentage of all possible observations of the variable are either less than 30 or greater than 40? (2 marks)
3. A curve has an area of 0.425 to the left of 4 and an area of 0.685 to the right of 4. Could this curve be a density curve for some variable? Explain your answer. (3 marks)
4. Determine the [latex]z[/latex]-scores [latex]z_{0.03}[/latex] and [latex]z_{0.005}[/latex]. (4 marks)
5. The finishing times in the New York City 10-km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes.
   1. Determine the percentage of finishers with times between 50 and 70 minutes. (4 marks)
   2. Determine the percentage of finishers with times less than 75 minutes. (3 marks)
   3. Obtain and interpret the 40th percentile for the finishing times. (4 marks: 3+1)
   4. Find and interpret the 8th decile for the finishing times. (4 marks: 3+1)

6. The weight of boxes of cereal follows a normal distribution with a mean of 1,000 grams and a standard deviation of 40 grams.
   1. The quality control department rejects a box if its weight is below 950 grams. What percentage of boxes will be rejected? (4 marks)
   2. Find the percentage of boxes weighing 980 grams and 1,010 grams. (4 marks)
   3. Find the percentage of boxes whose weight is above 1,010 grams. (4 marks)
   4. Determine the 40th percentile for the weight of the boxes of cereal. (4 marks)
   5. Determine the weight so that only 5% of boxes are heavier than the weight you choose. (4 marks)
   6. Randomly pick five boxes, find the probability that at least one box is rejected. (6 marks)

Part B

Finish the following questions using R and R commander. Make sure that you copy and paste the computer outputs into the space below each question, and write down your answers in statements.

1. Use R commander to find the [latex]z[/latex]-scores [latex]z_{0.03}[/latex] and [latex]z_{0.005}[/latex]. (4 marks)
2. The finishing times in the New York City 10-km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes.
   1. Determine the percentage of finishers with times between 50 and 70 minutes. (3 marks)
   2. Determine the percentage of finishers with times less than 75 minutes. (2 marks)
   3. Obtain the 40th percentile for the finishing times. (2 marks)
   4. Find the 8th decile for the finishing times. (2 marks)
3. The weight of boxes of cereal follows a normal distribution with a mean of 1,000 grams and a standard deviation of 40 grams.
   1. The quality control department rejects a box if its weight is below 950 grams. What percentage of boxes will be rejected? (2 marks)
   2. Find the percentage of boxes weighing 980 and 1,010 grams. (3 marks)
   3. Find the percentage of boxes whose weight is above 1,010 grams. (2 marks)
   4. Determine the 40th percentile for the weight of the boxes of cereal. (2 marks)
   5. Determine the weight such that only 5% of boxes are heavier than that. (2 marks)
   6. Randomly pick five boxes, find the probability that at least one box is rejected. (4 marks)
4. The monthly fees, in dollars, for a sample of the providers and plans are as follows: 40, 110, 90, 30, 70, 70, 30, 60, 60, 50, 60, 70, 35, 80, 75. Use R commander to assess whether the data follow a normal distribution. Is there any outlier? The data can be found in the file M05_MonthlyFees_labQ4.xlsx. (5 marks)