Deck 6: The Normal Distribution

Question 1

Which of the functions sketches below could be a probability density function for a continuous random variable?

A)
B)

Answer 1

Question 2

Determine the following probability from the curve f(x) diagrammed below.
P[1 < X < 2]

Answer 2

0.25

Question 3

For the probability distribution depicted below, determine:

A) median
B) first quartile
C) third quartile

Answer 3

Part A: 1
Part B: 0.5
Part C: 1.59

Question 4

Find the standardized variable Z if X has mean 27 and standard deviation 4.

Question 5

Find the standardized variable Z if X has mean 12 and variance 9.

Question 6

During a recent season, an NBA team from the Eastern Conference, had a mean of 108.33 points per game when scoring 100+, with a standard deviation of 2.92. Find the value of the standardized variable when X = 110. Round your answer to two decimal places.

A) 8.77
B) 0.57
C) 0.99
D) 0.96

Question 7

Assuming a normal distribution, determine the probability of the interval extending one sd on each side of the mean.

A)
B)
C)
D)

Question 8

Decide which statements are true about the curve of a normal distribution N( $\mu$ , $\sigma$ ).
I The curve is symmetric about its mean $\mu$ .
II The peak of the bell is located above the mean $\mu$ .
III The curve never reaches 0 for any value of x.
IV The graph shows the relation between different values of $\sigma$ .

A) Only I and II
B) Only I and III
C) Only II and III
D) All of them

Question 9

Find the area under the standard normal curve to the left of z = -2.97. Round your answer to four decimal places.

Question 10

Find the area under the standard normal curve to the right of z = -2.89. Round your answer to four decimal places.

Question 11

Find the area under the standard normal curve over the interval z = 1.36 to z = 1.53. Round your answer to four decimal places.

Question 12

For a standard normal random variable Z, find P[ Z < -1.79].

A) .0368
B) .0366
C) .0367
D) .0377

Question 13

For a standard normal random variable Z, find P[Z > 1.66].

A) .0485
B) .0486
C) .0487
D) .0484

Question 14

For a standard normal random variable Z, find P[ - .41 < Z < -.18].

A) .0876
B) .0877
C) .0878
D) .0879

Question 15

Given that P[ - z < Z < z ] = .9109, find the z value.

Question 16

A) Find P[ Z < .79 ].
B) Find the 79th percentile of the standard normal distribution. Round your answer to three decimal places.

Question 17

Records suggest that the normal distribution with mean 48 and standard deviation 9 is a plausible model for a measurement of the amount of people in a subway coach. Find
P[ X < 48.45 ].

Question 18

Records suggest that the normal distribution with mean 67 and standard deviation 11 is a plausible model for a measurement of the amount of people in a subway coach. Find P [ 42.25 $\le$ X $\le$ 77.67 ].

A) .822
B) .821
C) .818
D) .814

Question 19

Find b, given that P[X < b] = .695, where and σ=4.

Question 20

The Law School Admission Test (LSAT) is an integral part of law school admission. The collection of all scores in one year closely follows a normal distribution with mean 152 and a standard deviation of 11. Let this distribution apply to the applicant pool at
University W. Suppose the school decides to select only students who score above 160. What proportion of the applicants' pool would be eligible for selection?

Question 21

The Law School Admission Test (LSAT) is an integral part of law school admission. The collection of all scores in one year closely follows a normal distribution with mean 152 and a standard deviation of 11. Let this distribution apply to the applicant pool at
University W. What limit should the law school set that makes 40% of the applicants eligible?

Question 22

The acidity, pH, of rainfall in a region follows a normal distribution with mean 4.5 and standard deviation of .5. What should be the limit so that only most acidic (low pH) 15% of the rainfalls are subject to further chemical analysis? Round your answer to two decimal places.

Question 23

Let the number of successes X have a binomial distribution with n = 25 and p = 0.5. Find the normal approximation of P [ 11 $\le$ X $\le$ 14]. Apply the continuity correction

Question 24

According to U.S. Census Bureau, the percent of persons 65 years old and over, among those persons living in Virginia is 12.1%. Suppose you randomly select n = 330 persons in this state. Let success correspond to being part of this age group and let X denote the number of successes. Approximate the probability of X $\le$ 60. Round your answer to four decimal places.

Question 25

State whether or not the normal approximation to the binomial is appropriate in the following situation.
n = 315 and p = 0.17

Question 26

The median age of residents of Indiana is 35.2 years. If a survey of 300 residents is taken, approximate the probability that at least 125 will be under 35.2 years of age.
Apply the continuity correction.

Question 27

Suppose that 24% of the students in a large university are taking a mathematics course. What is the probability that in a random sample of 400 students, the number of students taking a mathematics course will be between 74 and 110 inclusive? Apply the continuity correction,

Question 28

1 Find the area under the standard normal curve
(a) to the left of z = .34,
(b) to the right of z = 1.27
Sketch the normal distribution and shade the area of interest.

Question 29

Find the area under the standard normal curve
(a) to the left of z = -1.61,
(b) to the right of z = .54
Sketch the normal distribution and shade the area of interest.

Question 30

Suppose the duration of trouble-free operation of a new robotic vacuum cleaner is normally distributed with mean 740 days and standard deviation 100 days.

(a) What is the probability that a new vacuum cleaner with be trouble-free for at least two years?
Sketch the normal distribution and shade the area of interest for the standard normal distribution.

(b) The manufacturer wishes to set the warranty period so that no more than 5 % of the robotic vacuum cleaners would require repair services while under warranty. Determine the warranty period.

Question 31

Suppose the amount of a sport drink in a bottle marked 100 milliliters is normally distributed with mean 101.6 ml and standard deviation
1.6 ml.

(a) What is the probability that the bottle you purchase contains less than 100 ml of the drink? two years?
Sketch the normal distribution and shade the area of interest for the standard normal distribution.
(b) If only 5 % of bottles contain less than b ml, what is the value of b?

Question 32

5 The data on volume of timber is measured on 49 randomly selected plots in a large forest. The normal scores plot of these data is below. Is the normal distribution plausible distribution for the volume of lumber in a plot?