Reference no: EM132231439
Assignment -
The Standard Normal Distribution
Q1. The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. What is the z-score for a patient who takes ten days to recover?
a. 1.5
b. 0.2
c. 2.2
d. 7.3
Q2. The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005-2006 season. The heights of basketball players have an approximate normal distribution with mean μ = 79 inches and a standard deviation σ = 3.89 inches. For each of the following heights, calculate the z-score and interpret it using complete sentences.
a. 77 inches
b. 85 inches
c. If an NBA player reported his height had a z-score of 3.5, would you believe him? Explain your answer.
Q3. Kyle's doctor told him that the z-score for his systolic blood pressure is 1.75. Which of the following is the best interpretation of this standardized score? The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with mean μ = 125 and standard deviation σ = 14. If X = a systolic blood pressure score then X ∼ N (125, 14).
a. Which answer(s) is/are correct?
i. Kyle's systolic blood pressure is 175.
ii. Kyle's systolic blood pressure is 1.75 times the average blood pressure of men his age.
iii. Kyle's systolic blood pressure is 1.75 above the average systolic blood pressure of men his age.
iv. Kyle's systolic blood pressure is 1.75 standard deviations above the average systolic blood pressure for men.
b. Calculate Kyle's blood pressure.
Q4. In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean μ = 520 and standard deviation σ = 115.
a. Calculate the z-score for an SAT score of 720. Interpret it using a complete sentence.
b. What math SAT score is 1.5 standard deviations above the mean? What can you say about this SAT score?
c. For 2012, the SAT math test had a mean of 514 and standard deviation 117. The ACT math test is an alternate to the SAT and is approximately normally distributed with mean 21 and standard deviation 5.3. If one person took the SAT math test and scored 700 and a second person took the ACT math test and scored 30, who did better with respect to the test they took?
Using the Normal Distribution
Q5. The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. The 90th percentile for recovery times is?
a. 8.89
b. 7.07
c. 7.99
d. 4.32
Q6. The length of time it takes to find a parking space at 9 A.M. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes. Find the probability that it takes at least eight minutes to find a parking space.
a. 0.0001
b. 0.9270
c. 0.1862
d. 0.0668
Q7. According to a study done by De Anza students, the height for Asian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Asian adult male is randomly chosen. Let X = height of the individual.
a. X ∼ _______(____,_____)
b. Find the probability that the person is between 65 and 69 inches. Include a sketch of the graph and write a probability statement.
c. Would you expect to meet many Asian adult males over 72 inches? Explain why or why not, and justify your answer numerically.
d. The middle 40% of heights fall between what two values? Sketch the graph and write probability statement.
Q8. The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of 10. Suppose that one individual is randomly chosen. Let X = percent of fat calories.
a. X ∼ _____(_____,_____)
b. Find the probability that the percent of fat calories a person consumes is more than 40. Graph the situation. Shade in the area to be determined.
c. Find the maximum number for the lower quarter of percent of fat calories. Sketch the graph and write the probability statement.
Q9. In China, four-year-olds average three hours a day unsupervised. Most of the unsupervised children live in rural areas, considered safe. Suppose that the standard deviation is 1.5 hours and the amount of time spent alone is normally distributed. We randomly select one Chinese four-year-old living in a rural area. We are interested in the amount of time the child spends alone per day.
a. In words, define the random variable X.
b. X ∼ _____(_____,_____)
c. Find the probability that the child spends less than one hour per day unsupervised. Sketch the graph, and write the probability statement.
d. What percent of the children spend over ten hours per day unsupervised?
e. Seventy percent of the children spend at least how long per day unsupervised?
Q10. Suppose that the duration of a particular type of criminal trial is known to be normally distributed with a mean of 21 days and a standard deviation of seven days.
a. In words, define the random variable X.
b. X ∼ _____(_____,_____)
c. If one of the trials is randomly chosen, find the probability that it lasted at least 24 days. Sketch the graph and write the probability statement.
d. Sixty percent of all trials of this type are completed within how many days?
Q11. Given Table shows a sample of the maximum capacity (maximum number of spectators) of sports stadiums. The table does not include horse-racing or motor-racing stadiums.
|
40,000
|
40,000
|
45,050
|
45,500
|
46,249
|
48,134
|
|
49,133
|
50,071
|
50,096
|
50,466
|
50,832
|
51,100
|
|
51,500
|
51,900
|
52,000
|
51,132
|
52,200
|
52,530
|
|
52,692
|
53,864
|
54,000
|
55,000
|
55,000
|
55,000
|
|
55,000
|
55,000
|
55,000
|
55,082
|
57,000
|
58,008
|
|
59,680
|
60,000
|
60,000
|
60,492
|
60,580
|
62,380
|
|
62,872
|
64,035
|
65,000
|
65,050
|
65,647
|
66,000
|
|
66,161
|
67,428
|
68,349
|
68,976
|
69,372
|
70,107
|
|
70,585
|
71,594
|
72,000
|
72,922
|
73,379
|
74,500
|
|
78,025
|
76,212
|
78,000
|
80,000
|
80,000
|
82,300
|
a. Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums (the data).
b. Construct a histogram.
c. Draw a smooth curve through the midpoints of the tops of the bars of the histogram.
d. In words, describe the shape of your histogram and smooth curve.
e. Let the sample mean approximate μ and the sample standard deviation approximate σ. The distribution of X can then be approximated by X ∼ _____(_____,_____).
f. Use the distribution in part e to calculate the probability that the maximum capacity of sports stadiums is less than 67,000 spectators.
g. Determine the cumulative relative frequency that the maximum capacity of sports stadiums is less than 67,000 spectators. Hint: Order the data and count the sports stadiums that have a maximum capacity less than 67,000. Divide by the total number of sports stadiums in the sample.
h. Why aren't the answers to part f and part g exactly the same?
Q12. A NUMMI assembly line, which has been operating since 1984, has built an average of 6,000 cars and trucks a week. Generally, 10% of the cars were defective coming off the assembly line. Suppose we draw a random sample of n = 100 cars. Let X represent the number of defective cars in the sample. What can we say about X in regard to the 68-95-99.7 empirical rule (one standard deviation, two standard deviations and three standard deviations from the mean are being referred to)? Assume a normal distribution for the defective cars in the sample.
Q13. A $1 scratch off lotto ticket will be a winner one out of five times. Out of a shipment of n = 190 lotto tickets, find the probability for the lotto tickets that there are
a. somewhere between 34 and 54 prizes.
b. somewhere between 54 and 64 prizes.
c. more than 64 prizes.