Question

1. The Zumber National Rent Report lists the average monthly apartment rent in various locations in the United States. According to their report, the average cost of renting a one-bedroom apartment in Houston is $1,090. Suppose that the standard deviation of the cost of renting a one-bedroom apartment in Houston is $96 and that such apartment rents in Houston are normally distributed. If a one-bedroom apartment in Houston is randomly selected, what is the probability that the price is: (20 points) (a) More than $1,250? (b) Between $1,050 and $1,200? (c) Between $900 and $1,000? (d) Less than $1,180? Bonus question (e) Below $850 or beyond $1,300? (5 points)

2. Suppose the grade on a Math test is normally distributed with mean 78 and standard deviation 10. (a) Compute the z-scores (5 points) (a-1) If Bob got 70 on the test, what is his z-score? (a-2) If Jane got 90 on the test, what is her z-score? (b) Compute the actual grades (5 points) (b-1) Suppose David achieved a grade 1.8 standard deviation above the mean (? = 1.8), what was his actual grade? (b-2) Suppose Lily achieved a grade 0.5 standard deviation below the mean (? = −0.5), what was her actual grade? (c) Rob achieved a grade that exceeded 95% of all grades. Find Rob’s actual grade. (6 points) (d) Suppose 32% of students did better than Mei. Find Mei’s actual grade. (6 points)

3. According to the U.S. Department of Agriculture, Alabama egg farmers produce millions of eggs every year. Suppose egg production per year in Alabama is normally distributed, with a standard deviation of 83 million eggs. (a) If during only 3% of the years Alabama egg farmers produce more than 2,655 million eggs, what is the mean egg production by Alabama farmers? (6 points) (b) Given the mean egg production in (a), what is the probability that Alabama egg farmers produces less than 1,500 million eggs? (5 points) (c) Given the mean egg production in (a), what is the probability that Alabama egg farmers produces eggs between 2,000 and 2,500 million eggs? (5 points)

4. Suppose the speeds of people driving from Dallas to Houston normally distributed, with a mean average speed of 92 miles per hour. (a) If the probability that the speed is more than 108 mph is 0.8%, what is the standard deviation? (6 points) (b) Given the standard deviation in (a), what is the probability that a person drove less than 85 miles per hour? (5 points) (c) If Ben drove at a speed that exceeds 85% of all other people. What was her speed? (6 points)

5. According to the U.S. Census Bureau, 20% of the workers in Atlanta use public transportation. Suppose 25 Atlanta workers are randomly selected. (Hint: use sampling distribution) (a) What is the standard deviation of the sample proportion of the selected workers who use public transportation? (5 points) (a) What is the probability that the proportion of the selected workers who use public transportation is less than 32%? (5 points) (b) What is the probability that the proportion of the selected workers who use public transportation is greater than 48%? (5 points)

6. Gross weights of 8-ounce boxes of cereal are normally distributed with mean 9.60 ounces and standard deviation 0.80 ounces. Boxes are packaged 24 per carton. Find the probability of randomly selected 24 boxes having average weight between 9.5 and 10 ounces. (Hint: use sampling distribution) (10 points)

Answer 1

1:

Here we have

(a)

The z-score for X = 1250 is

The probability that the price is More than $1,250 is

P(X > 1250) = P(z > 1.67)=0.0475

(b)

The z-score for X = 1050 is

The z-score for X = 1200 is

The probability that the price is Between $1,050 and $1,200 is

P(1050<= X <= 1200) = P(-0.42<= z <= 1.15)=P(z <= 1.15) - P(z <= -0.42) = 0.5377

(c)

The z-score for X = 900 is

The z-score for X = 1000 is

The probability that the price is Between $900 and $1000 is

P(900<= X <= 1000) = P(-1.98<= z <= -0.94)=P(z <= -0.94) - P(z <= -1.98) = 0.1498

(d)

The z-score for X = 1180 is

The probability that the price is Less than $1,180 is

P( X < 1180) = P(z <= 0.94)=0.8264

(e)

The z-score for X = 850 is

The z-score for X = 1300 is

The probability that the price is Below $850 or beyond $1,300 is

P( X < 850) +P(X > 1300) = P(z < -2.5) +P(z > 2.19)=0.0062+0.0143=0.0205