Normal and Sampling Distribution

Answer the following questions. Show the complete and legible answers/solutions.Use 4 decimal places for the finals answers.

Question 3

A prototype automotive tire has a design life of 38,500 miles with a standard deviation of 2,500 miles. Five such tires are manufactured and tested. On the assumption that the actual population mean is 38,500 miles and the actual population standard deviation is 2,500 miles, find the probability that the sample mean will be less than 35,000 miles. Assume that the distribution of lifetimes of such tires is normal.

Question 4

A normally distributed population has mean 1,200 and standard deviation 120.

a.) Find the probability that a single randomly selected element X of the population is between 1,100 and 1,300.

b.) Find the mean and standard deviation of X for samples of size 25.

c.) Find the probability that the mean of a sample of size 25 drawn from this population is between 1,100 and 1,300.

Question 5

Scores on an entrance exam in a large university, freshman course are normally distributed with mean 72.5 and standard deviation 13.0.

a.) Find the probability that the score X on a randomly selected exam paper is between 70 and 80.

b.) Find the probability that the mean score X of 40 randomly selected exam papers is between 70 and 80.

Question 6

Suppose the mean weight of school children's book bags is 17.5 pounds, with standard deviation 2.2 pounds. Find the probability that the mean weight of a sample of 30 book bags will exceed 18 pounds.

Question 7

The mean and standard deviation of the tax value of all vehicles registered in NCR are =550,000 and =80,000. Suppose random samples of size 100 are drawn from the population of vehicles. What are the mean x and standard deviation x of the sample mean X?

Question 13

The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under guarantee. If he is willing to replace only 5% of the motors that fail, how long a guarantee should he offer? Assume that the lifetime of a motor follows a normal distribution.

Question 14

The mean IQ of 320 college students of a certain university is 100 with a standard deviation of 15. Assuming that IQ's are normally distributed; find the number of students with IQ

a.) above 140

b.) below 80

c.) What percent of the cases cannot score above 110?

Question 15

A. Find the area under the standard normal curve.

1. from z = 0 to z = 1.46
2. from z = -0.32 to z = 0.98
3. from z = 0.07 to z = 2.51
4. to the right of z = 2.13
5. to the left of z = 1.04

B. Find the value of z so that the area under the standard normal curve

1. from 0 to z is (approximately) 0.1965 and z is positive
2. between 0 and z is (approximately) 0.2740 and z is negative
3. in the left tail is (approximately) 0.2050
4. to the right of z is (approximately) 0.6285