stat

STAT 350 (Spring 2016) Homework 5 (20 points + 3 points BONUS) 1

(1 pt.) 1. Examine each of the normal probability plots below. Is there any evidence to suggest the data are from a normal population? If it is from a non-normal population, which population is it from? Justify your answer. In both of these plots, the y-axis is the observed values and the x-axis is the theoretical values.

(0.5 pts.) a)

(0.5 pts.) b)

Additional Problems: 6.81 (1 pt. + 1 pt. Bonus) 2. If X is a uniform distribution defined over the interval (a,b) verify that

(1 𝑝𝑡. ) 𝑎) 𝐸(𝑋) = 𝑎 + 𝑏

2

(1 𝑝𝑡. 𝐵𝑂𝑁𝑈𝑆) 𝑏) 𝑉𝑎𝑟(𝑋) = (𝑏 − 𝑎)2

12 (𝐻𝑖𝑛𝑡: 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑐𝑜𝑚𝑝𝑢𝑡𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎)

(2 pts.) 3 Pre-manufactured wooden roof trusses allow builders to complete projects faster and

with lower on-site labor costs. The connector plates for trusses are made from Grade A steel and are hot-dip galvanized. The thickness of a truss connector (in inches) varies slightly and has a uniform distribution with from 0.036 inches to 0.050 inches. Please provide 3 decimal places for this problem.

(1 pt.) a) If the manufacturer will only use connectors with a minimum thickness of 0.04 inch,

what proportion of connectors is rejected? (0.5 pts.) b) Find the mean thickness of the truss connectors. (0.5 pts.) c) Find the standard deviation of the thickness of the truss connector. Additional Problems: 6.13, 6.15, Practice Problems: 6.85 (p. 287)

(1 pt.) 4. Derive the cdf for an exponential distribution with parameter .

STAT 350 (Spring 2016) Homework 5 (20 points + 3 points BONUS) 2

(2 pt. Bonus) 5. If X is an exponential distribution with parameter , verify that

(1 𝑝𝑡. 𝐵𝑂𝑁𝑈𝑆) 𝑎) 𝐸(𝑋) = 1

𝜆 .

(1 𝑝𝑡. 𝐵𝑂𝑁𝑈𝑆) 𝑏) 𝑉𝑎𝑟(𝑋) = 1

𝜆2

(3 pts.) 6 A toy manufacturer routinely tests new toys in a controlled environment before

deciding whether to actually market a toy. Research has shown that the amount of time (in minutes) a randomly selected child plays with a new toy has an exponential distribution with an average of 20 minutes. Suppose a new toy is presented for study.

(0.5 pt.) a) Find the value of λ. (0.5 pt.) b) What is the standard deviation of the length of time that a child will play with the toy? (1 pt.) c) If the child plays with a toy for more than 30 minutes, it will have to be cleaned before

the next child can play with it. What is the probability that the toy will have to be cleaned? (1 pt.) d) Find the probability that the child will play with the toy between 5 and 21 minutes. Additional Problems: 6.93, 6.95, 6.97 Practice Problems: 7.1 (p.301), 7.3 (p.301) (1.5 pts) 7. In each of the following statements, identify the boldface number as the value of a

population parameter or a sample statistic. You do not need to explain your answer. (0.5 pts.) a) A toy manufacturer keeps records on all sales of all toys records of all returns. The

manufacturer issued a recall on a small wooden toy car because the wheels could break off and pose a choking hazard for small children. The proportion of buyers who took advantage of the recall was reported to be 0.45.

(0.5 pts.) b) In a random sample of dentists, 80% recommended a certain product to help whiten teeth.

(0.5 pts.) c) During a recent winter, the month of January was particularly cold in Pennsylvania. One power company in the state reported that the mean number of kilowatts used by each customer during this month was 1346.

Additional Problems: 7.11 Practice Problems: 7.27 (p.314), 7.29 (p.314), 7.31 (p.314)

STAT 350 (Spring 2016) Homework 5 (20 points + 3 points BONUS) 3

(1 pt.) 8. The figure below shows graphs of the probability density function for the random variable X and the approximate density functions for the random variable for n= 5 and the random variable for n = 15. Identify which probability density function corresponds to which random variable. Please explain your answer.

Additional Problems: 7.37 (3.5 pts.) 9. This is similar to Example 7.6 (Section 7.2). Also look at Example 7.3 (Section 7.1).

For planning purposes, U.S. Bancorp has determined the probability distribution for the retirement age, X, of employees in the mortgage work group. The probability distribution for X is given in the table below.

x 65 66 67

p( x) 0.1 0.5 0.4

(0.5 pts.) a. Find the mean of X. (2 pts.) b. Suppose two employees from this work group are selected at random. Find the exact

probability distribution for the sample mean, X̄. Hint: Intersection. (0.5 pts.) c. Find the mean of X̄. How does this compare with your answer in part (a)? (0.5 pts.) d) Does this confirm or deny the Properties of the Sample Mean on p. 306 (Section

7.2) in the textbook. Please explain your answer. (3 pts.) 10. One measure of general health is your body mass index (BMI). Adults with a BMI

between 18.5 and 24.9 are generally considered to have an ideal body weight for their height. Despite a reputation as a food-loving nation, the mean BMI for adults in France is 24.5. Suppose the BMI for adults in France is normally distributed with standard deviation 1.3.

(1 pt.) a) Suppose one adult from France is selected at random. What is the probability that the

person’s BMI is more than 25? (1 pt.) b) Suppose 10 adults from France are selected at random. What is the probability that the

sample mean BMI is greater than 25? (1 pt.) c) Why are the answers to parts a) and b) different? In particular, which probability is

smaller, and explain the theory for why this probability is smaller (without referencing the calculations).

STAT 350 (Spring 2016) Homework 5 (20 points + 3 points BONUS) 4

(3 pts.) 11. The ozone hole is a region in the atmosphere in the Southern Hemisphere mainly over Antarctica. The mean ozone hole area (in million km2) in 2012 was 17.9 with a standard deviation of 7.4. Suppose 60 days are selected at random and the ozone hole is measured each day.

(1 pt.) a) What is the probability that the sample mean ozone hole area is less than 15.5 million

km2? (1 pt.) b) What is the probability that the sample mean ozone hole area is between 16 and 18

million km2? (1 pt.) c) Some researchers suggest that increased use of chemicals has caused the ozone hole

area to increase. Suppose the sample mean for 60 days selected at random in 2016 is 20.7 million km2. Is there any evidence to suggest that the mean ozone hole area has increased since 2012? Justify your answer. Hint: What is the probability that you obtain this level or higher when using the original distribution from 2012?

Additional Problems: 7.41, 7.43, 7.45, 7.47, 7.49