Deck 6: The Normal Probability Distribution

Question 1

In general, the binomial probability P(X x) is approximated by the area under the normal curve to the right of x - .5.

Answer 1

True

Question 2

The total area under any continuous probability curve is equal to one.

Answer 2

True

Question 3

Adding or subtracting 0.5 from the binomial random variable x's interval endpoints (i.e., the so-called continuity correction) improves the normal approximation to the binomial by correcting for or accounting for the "missing" corners of the probability rectangles corresponding to values of x.

Answer 3

True

Question 4

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if:

A) the sample size n is greater than 30
B) the sample size n is greater than 100
C) the population proportion p is close to 0.50
D) the underlying population is normal
E) np and n(1 - p) are both greater than or equal to 5

Question 5

Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the area under a normal curve to the right of:

A) x - 0.5
B) x + 0.5
C) x - 1
D) x + 1
E) x + 0.1

Question 6

Suppose that the probability p of success on any trail of a binomial distribution equals 0.90. Then for which of the following number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?

A) 25
B) 35
C) 45
D) 55
E) 65

Question 7

Twenty-five psychology instructors have formed a committee to pick next year's textbook, and they have narrowed their decision down to two equally good books, one with a better bibliography and references, and the other with a better format and illustrations. Since the books are considered to be equally good, we will assume the probability an instructor chooses either book is 0.5 and the instructors' decisions are made independently.
Using the binomial distribution, find the probability 15 or more instructors choose the book with the better format and illustrations.
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Use the normal approximation to the binomial to find the probability 15 or more instructors choose the book with the better format and illustrations.
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Compare the results in parts (a) and (b). Is the normal approximation a good approximation?
______________

Question 8

In general, the binomial probability P(X = x) is approximated by the area under a normal curve between x - .5 and x + .5.

Question 9

Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the area under a normal curve to the left of:

A) x
B) -x
C) x + 0.5
D) x - 0.1
E) -x - 0.1

Question 10

Consider a binomial random variable x with n = 25 and p = 0.01. Would the normal approximation be appropriate here?
______________
Why or why not?
________________________________________________________

Question 11

Past records indicate 60% of the eligible voters in Michigan vote in the primary election.
What is the expected number in a random sample of 200 eligible voters who will vote in the next primary?
______________
What is the approximate probability more than 150 from a sample of 200 will vote in the next primary?
______________

Question 12

The normal approximation to the binomial distribution works best when the number of trials is large, and when the binomial distribution is symmetrical (like the normal).

Question 13

A recent survey of university faculty reported 55% were considering going into another profession. Based on this information, what is the expected number of faculty who would be considering going into another profession if we randomly sampled 200 faculty members?
______________
What is the approximate probability 60 or more faculty from a random sample of 200 would be considering going into another profession?
______________

Question 14

All binomial distributions can be approximated very closely by normal distributions.

Question 15

Even though the normal probability distribution deals with continuous variables, we can use it to approximate the binomial whenever np > 5 and also nq > 5.

Question 16

Even though the normal probability distribution deals with continuous variables, we can use it to approximate the binomial probability distribution whenever n 30 and also n(q) 30.

Question 17

Given that X is a binomial random variable, the binomial probability P(X = x) is approximated by the area under a normal curve between:

A) x - 0.5 and 0.0
B) 0.0 and x + 0.5
C) 1 - x and 1 + x
D) x - 0.5 and x + 0.5
E) 0.0 and x + 1

Question 18

The normal approximation to the binomial is appropriate when the number of trials n is large and the probability of success p is near 0.05.

Question 19

In general, the binomial probability P(X x) is approximated by the area under the normal curve to the left of x + - .5.

Question 20

Even though the normal probability distribution deals with continuous variables, under certain circumstances, we can use it to approximate various discrete probability distributions. Such use of the normal probability distribution is permissible:

A) in the case of binomial probabilities, when np > 5 and also nq > 5
B) in the case of Poisson probabilities, when> 100
C) in the case of hypergeometric probabilities, when np > 1 and also n1 > 1
D) in all of these
E) in none of these

Question 21

A normal random variable x is standardized by expressing its value as the number standard deviations it lies to the right or left of its mean .

Question 22

Suppose the records of a motel show that, on the average, 10% of their prospective guests will not claim their reservation. If the motel accepts 218 reservations and there are only 200 rooms in the motel, what is the probability that all guests who arrive to claim a room will receive one?
______________

Question 23

Any continuous random variable cannot take on negative values.

Question 24

A student government representative claims that 55% of the student body of a local university favor a move to Division I in college football. A random sample of 2000 students is selected.
What is the expected number of students that will favor the move to Division I?
______________
What is the approximate probability that less than 1150 students of the 2000 selected will favor the move to Division I?
______________

Question 25

For a continuous random variable x, the probabilities P(0 x 1) and P(0 < x < 1) are always equal.

Question 26

Let x be a binomial random variable with n = 25 and p = 0.3.
Is the normal approximation appropriate for this binomial random variable?
______________
Find the mean for x.
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Find the standard deviation for x.
______________
Use the normal approximation to find .
______________
Use the cumulative binomial probabilities table in Appendix 1 of your book to calculate the exact probability, above.
______________
Compare the results of the last two questions. Was your approximation close, or far, from the exact answer?
______________

Question 27

The relative frequency associated with a particular class in the population is the fraction of measurements in the population falling in that class.

Question 28

Historical data collected at First of America Bank in Michigan revealed that 80% of all customers applying for a loan are accepted. Suppose that 50 new loan applications are selected at random.
Find the expected value of the number of loans that will be accepted by the bank.
______________
Find the standard deviation of the number of loans that will be accepted by the bank.
______________
What is the probability that at least 42 loans will be accepted?
______________
What is the probability that the number of loans rejected is between 10 and 15, inclusive?
______________

Question 29

The total area under the curve f(x) of any continuous random variable x is equal to one.

Question 30

In a certain population, 18% of the people have Rh-negative blood. A blood bank serving this population receives 95 blood donors on a particular day.
What is the probability that 10 or fewer are Rh-negative?
______________
What is the probability that 15 to 20 (inclusive) of the donors are Rh-negative?
______________
What is the probability that more than 80 of the donors are Rh-positive?
______________

Question 31

Suppose the current median age of U.S. citizens is 30. If a survey of 500 randomly selected U.S. citizens is conducted, what is the probability at most 240 of them will be under 30 years old?
______________

Question 32

Do Americans tend to vote for the taller of the two candidates in a presidential election? In 30 of our presidential elections since 1856, 18 of the winners were taller than their opponents. Assume that Americans are not biased by a candidate's height and that the winner is just as likely to be taller or shorter than his opponent. Is the observed number of taller winners in the U.S. presidential elections unusual?
Find the approximate probability of finding 18 or more of the 30 pairs in which the taller candidate wins.
______________
Based on your answer to part (a), can you conclude that Americans consider a candidate's height when casting their ballot?
______________

Question 33

Continuous random variables can assume infinitely many values corresponding to points on a line interval.

Question 34

Twenty percent of American men over 30 have high blood pressure. To run an experiment, a researcher needs to identify at least 15 men with high blood pressure. If the researcher examines 100 men over 30, what is the approximate probability he or she finds enough men with high blood pressure to run the experiment?
______________

Question 35

A firm with many traveling salespersons decides to check the salespersons' travel expenses to see if they are correctly reported. An auditor for the firm selects 200 expense reports at random to audit. What is the probability more than 20% of the sampled reports will be incorrect when, in fact, only 10% of the firm's expense reports are improperly documented?
______________

Question 36

For any continuous random variable x, P(x = 5) = 1 is always true.

Question 37

For a continuous random variable x, P(6 < x < 8) is represented by the area under the probability function over the interval from 6 to 8.

Question 38

The relative frequency associated with a particular class in the population is the probability of drawing a measurement from that class.

Question 39

The admissions office of a community college in Indiana is asked to accept deposits from a number of qualified prospective freshmen so that, with probability about 0.95, the size of the freshmen class will be less than or equal to 150. Suppose the applicants constitute a random sample from a population of applicants, 80% of whom would actually enter the freshmen class if accepted.
How many deposits should the admissions counselor accept?
______________
If applicants in the number determined in part (a) are accepted, what is the probability that the freshmen class size will be less than 135?
______________

Question 40

The mean, median, and standard deviation of a normally distributed random variable are all at the same position on the horizontal axis since the distribution is symmetric.

Question 41

Let be a z-score that is unknown but identifiable by position and area. If the symmetrical area between - and + is 0.903, the value of is 1.66.

Question 42

If x is a normal random variable with mean of 2 and standard deviation of 5, then P(x < 3) = P(x > 7).

Question 43

The z-score representing the 10th percentile of the standard normal curve is -1.28.

Question 44

Given a normal random variable x with mean and standard deviation , the mean of the standard normal random variable z associated with x is 1.

Question 45

The proportion of the total area under the normal curve that lies within one standard deviation of the mean is approximately 0.75.

Question 46

Using the standard normal curve, the area between z = -1.50 and z = 1.50 is 0.4332.

Question 47

Using the standard normal curve, the area between z = 0 and z = 2.2 is 0.4868.

Question 48

Given a normal random variable x with mean and standard deviation , the standard normal random variable associated with x is .

Question 49

Given that z is a standard normal random variable, a negative value of z indicates that the standard deviation of z is negative.

Question 50

Let be a z-score that is unknown but identifiable by position and area. If the area to the right of is 0.8849, the value of is 1.2.

Question 51

Given that z is a standard normal random variable, a value of z that is equal to zero indicates that the standard deviation of z is zero also.

Question 52

Given a normal random variable x with mean and standard deviation , the variance of the standard normal random variable z associated with x is 0.

Question 53

The shape of the normal probability distribution is determined by the population standard deviation . Small values of reduce the height of the curve and increase the spread; large values of increase the height of the curve and reduce the spread.

Question 54

A continuous random variable x is normally distributed with a mean of 1200 and a standard deviation of 150. Given that x = 1410, its corresponding z-score is 1.40.

Question 55

Given a normal random variable x with mean of 82 and standard deviation of 12, the value of the standard normal random variable z associated with x = 70 is smaller than zero.

Question 56

Using the standard normal curve, the z-score representing the 95th percentile is 1.645.

Question 57

The left half of the normal curve is slightly smaller than the right half.

Question 58

If x is a normal random variable with mean = 4 and standard deviation = 2, and Y is a normal random variable with mean = 10 and standard deviation = 5, then P(x < 0) = P(Y < 0).

Question 59

The normal probability distribution is important because a large number of random variables observed in nature possess a frequency distribution that is approximately a normal probability distribution.

Question 60

Given a normal random variable x with mean of 70 and standard deviation of 12, the value of the standard normal random variable z associated with x = 82 is larger than zero.

Question 61

A continuous random variable x is normally distributed with a mean of 100 oz. and a standard deviation of 20 oz. Given that x = 120, its corresponding z-score is -1.0.

Question 62

The normal random variable's density function is (1) single-peaked above the variable's mean, median, and mode, all of which are equal to one another, (2) perfectly symmetric about this peaked central value and, thus, bell-shaped, and (3) characterized by tails extending indefinitely in both directions from the center, approaching (but never touching) the horizontal axis, which implies a positive probability for finding values of the random variable anywhere between minus infinity and plus infinity.

Question 63

The mean and variance of any normal distribution are always zero and one, respectively.

Question 64

For a continuous random variable x, P(x a) = P(x > a) and P(x a) = P(x < a).

Question 65

A standard normal curve is a normal probability density function with a mean of 1 and a standard deviation of 0.

Question 66

A smooth frequency curve that describes the probability distribution of a continuous random variable is called a standard normal curve.

Question 67

For any random variable, P(x a) = P(x < a) because P(x = a) = 0.

Question 68

One difference between a binomial random variable x and a standard normal random variable z is that with the binomial random variable we cannot determine P(x = a), while for the standard normal variable we can determine P(z = a).

Question 69

The z-score representing the 99th percentile of the standard normal curve is 3.0.

Question 70

The normal probability distribution is right-skewed for large values of the standard deviation , and is left-skewed for small values of .

Question 71

For any random variable x, P(x = a) = 0.

Question 72

The normal probability distribution is one of the most commonly used discrete probability distributions.

Question 73

Assume that x is a normally distributed random variable with a mean equal to 13.4 and a standard deviation equal to 3.6. Given this information and that P(x > a) = .05, the value of a must be 19.322.

Question 74

For a continuous random variable, P(x a) = P(x > a) because P(x = a) = 0.

Question 75

Continuous random variables can assume values at all points on an interval, with no breaks between possible values.

Question 76

The possible observations about continuous random variables are infinite in number; characteristics measured in units of money, time, distance, height, volume, length, or weight exemplify the matter.

Question 77

Examples of continuous probability distributions include the normal probability distribution, the Poisson probability distribution, and the binomial probability distribution.

Question 78

The total area under the normal probability distribution curve is equal to 1.

Question 79

If the mean, median, and mode are all equal for a continuous random variable, then the random variable must be normally distributed.

Question 80

The time it takes a student to finish a final exam test is known to be normally distributed with a mean equal to 84 minutes with a standard deviation equal to 10 minutes. Given this information, the probability that it will take a randomly selected student between 75 and 90 minutes is approximately .0902.