ST 560 Module 6

Z is a standard normal random variable. Compute P(Z > -1.75). a.0.96 b.0.93 c.0.04 d.0.07

a.0.96

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. How many steps would he have to take to make the cut for the top 5% for his distribution. a.12,467 b.10,000 c.7,533 d.8,078

a.12,467

The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. What is the variance of this distribution? a.12.25 minutes b.7 minutes c.1.87 minutes d.3.5 minutes

a.12.25 minutes For exponential distribution, µ = 𝜎 𝜎 = 3.5 𝜎² = 3.5² = 12.15

Suppose that a basketball player scored, on average, 15 points per game. Also suppose that the distribution of points scored by this player was normal. If he scores 20 points or more 4.78% of the time, what is his standard deviation? a.3 b.2.5 c.1.5 d.2

a.3 Z-score for 95.22nd percentile is 1.66 z = (x - µ) / 𝜎 𝜎 = (x - µ) / z = (20-15)/1.66 = 5/1.66 = 3.01

The random variable X is known to be uniformly distributed between 2 and 12. Compute E(X). a.7 b.6 c.4 d.5

a.7

A binomial probability distribution has p = 0.25 and n = 80. Using the normal approximation to the binomial distribution, what is probability of 25 or fewer successes? a.90% b.80% c.60% d.70%

a.90%

Which of the following statements is correct? a.The binomial distribution is a discrete probability distribution and the normal distribution is a continuous probability distribution. b.The binomial distribution is a continuous probability distribution and the normal distribution is a discrete probability distribution. c.The binomial and normal distributions are both discrete probability distributions. d.The binomial and normal distributions are both continuous probability distributions.

a.The binomial distribution is a discrete probability distribution and the normal distribution is a continuous probability distribution.

Which of the following is not a characteristic of the normal probability distribution? a.The standard deviation must be 1. b.The distribution is symmetrical. c.The mean, median, and the mode are equal. d.The mean of the distribution can be negative, zero, or positive.

a.The standard deviation must be 1.

If arrivals follow a Poisson probability distribution, the time between successive arrivals must follow a.an exponential probability distribution. b.a uniform probability distribution. c.a Poisson probability distribution. d.a normal probability distribution.

a.an exponential probability distribution.

Approximate the following binomial probabilities by the use of normal approximation. In a national poll 80% of adults responded that they typically watch TV in the evening. What is the probability that in a random sample of 100 adults between 70-80 people say that they watch TV in the evening? a.0.69 b.0.49 c.0.09 d.0.29

b.0.49 Mean = 80 and SD = 4; therefore, 70 successes correspond to a z-score of -2.5 and 80 correspond to a z-score of 0. P(-2.5 ≤ z ≤ 0) = 0.49

Z is a standard normal random variable. Compute P(-1.5 < Z < 1.5). a.1 b.0.87 c.0 d.0.43

b.0.87

Approximate the following binomial probabilities by the use of normal approximation. In a national poll 80% of adults responded that they typically watch TV in the evening. What is the probability that in a random sample of 100 adults less than 85 people say that they watch TV in the evening? a.0.93 b.0.89 c.0.75 d.0.80

b.0.89 The mean is 80 and the SD is 4, therefore, 85 successes correspond to a z-score of 1.25. P(z < 1.25) = 0.89

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. One day he took 15,000 steps. What was his percentile on that day? a.100% b.99.7% c.95% d.97.7%

b.99.7%

A normal distribution with a mean of 0 and a standard deviation of 1 is called a.none of these alternatives is correct. b.a standard normal distribution. c.a probability density function. d.an ordinary normal curve.

b.a standard normal distribution.

A value of 0.5 that is added and/or subtracted from a value of x when the continuous normal distribution is used to approximate the discrete binomial distribution is called a a.discrete probability conversion. b.continuity correction factor. c.a factor of conversion. d.none of the above are correct.

b.continuity correction factor.

A continuous probability distribution that is useful in describing the time, or space, between occurrences of an event is a(n) a.Poisson probability distribution. b.exponential probability distribution. c.normal probability distribution. d.uniform probability distribution.

b.exponential probability distribution.

There is a lower limit but no upper limit for a random variable that follows the a.binomial probability distribution. b.exponential probability distribution. c.uniform probability distribution. d.normal probability distribution.

b.exponential probability distribution.

The random variable X is known to be uniformly distributed between 2 and 12. Compute P(X = 6). a.There is not enough information to compute this probability. b.3/5 c.0 d.1/6

c.0 A single point is an interval of zero width. This implies that the probability of a continuous random variable assuming any particular value exactly is zero.

A binomial probability distribution has p = 0.25 and n = 80. Using the normal approximation to the binomial distribution, what is probability of obtaining more than 30 successes? a.0.9951 b.0.7450 c.0.0049 d.0.3561

c.0.0049

Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that The time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the probability that it takes between 2-3 minutes to fill an order? a.0.3133 b.0.2755 c.0.1283 d.0.2423

c.0.1283

The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. What is the probability that it takes more than 5 minutes to ring up a customer? a.0.7603 b.0.6321 c.0.2397 d.0.3679

c.0.2397 Cumulative probability P(X≤5) = 1 - e^(-5/3.5) = 0.7603 so P(X>5) = 1 - 0.7603 = 0.2397

The height of the probability density function for a uniform distribution ranging between 2 and 6 is a.4. b.1. c.0.25. d.0.5.

c.0.25.

The newest model of smart car is supposed to get excellent gas mileage. A thorough study showed that gas mileage (measured in miles per gallon) is normally distributed with a mean of 75 miles per gallon and a standard deviation of 10 miles per gallon. What is the probability that, if driven normally, the car will get 100 miles per gallon? a.25% b.6% c.0.6% d.2.5%

c.0.6%

The random variable X is known to be uniformly distributed between 2 and 12. Compute the standard deviation of X. a.12 b.8.333 c.2.887 d.3.464

c.2.887 𝜎 = √((b-a)²/12) = √((12 - 2)²/12) = √8.33 = 2.887

A binomial probability distribution has p = 0.25 and n = 80. What is the standard deviation of this distribution? a.8.94 b.7.75 c.3.87 d.4.47

c.3.87 = √np(1-p) = √(20)(.75) = 3.87

Approximate the following binomial probabilities by the use of normal approximation. In a national poll 80% of adults responded that they typically watch TV in the evening. What is the probability that in a random sample of 100 adults 85 people say that they watch TV in the evening? a.34.6% b.24.6% c.4.6% d.14.6%

c.4.6%

The newest model of smart car is supposed to get excellent gas mileage. A thorough study showed that gas mileage (measured in miles per gallon) is normally distributed with a mean of 75 miles per gallon and a standard deviation of 10 miles per gallon. What value represents the 50th percentile of this distribution? a.105 b.95 c.75 d.85

c.75

Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that The time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the probability that it takes less than one minute to fill an order? a.0.7769 b.0.6321 c.0.1813 d.0.4866

d.0.4866

For the standard normal probability distribution, the area to the left of the mean is a.1. b.any value between 0 to 1. c.-0.5. d.0.5.

d.0.5.

Let the random variable represent the profit made on a randomly selected day by a certain store. Assume is normally distributed with a mean of $360 and standard deviation of $50. What is the -score that corresponds to the likelihood the profit is $400? a.1.0 b.0.9 c.0.7 d.0.8

d.0.8

A binomial probability distribution has p = 0.25 and n = 80. Using the normal approximation to the binomial distribution, what is probability of obtaining 20 successes? a.20.5% b.15.4% c.5.2% d.10.3%

d.10.3% Use the continuity correction factor to calculate this probability. Find P(19.5 ≤ X ≤ 20.5). This equates to P(-0.13 ≤ z ≤ 0.13) = 10.3%

Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that The time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the variance of this distribution? a.Cannot be determined from the information given. b.3 minutes c.1.5 minutes d.2.25 minutes

d.2.25 minutes For exponential distribution, µ = 𝜎 𝜎 = 1.5 𝜎² = 1.5² = 2.25

What is the mean of x, given the exponential probability function f(x)= (1/20) e^(-x/20) for x≥0? a.2,000 b.100 c.0.05 d.20

d.20

In a standard normal distribution, what Z-score corresponds to the 75th percentile? a.Z = 1.75 b.Z = 0.75 c.Z = 0.07 d.Z = 0.67

d.Z = 0.67

When np ≥ 5 and n(1 − p) ≥ 5, the binomial probabilities can be approximated by a a.exponential distribution. b.discrete probability distribution. c.uniform distribution. d.normal distribution.

d.normal distribution.

For a uniform probability density function, a.the height of the function decreases as x increases. b.the height of the function is different for various values of x. c.the height of the function cannot be larger than one. d.the height of the function is the same for each value of x.

d.the height of the function is the same for each value of x.

The center of a normal curve is a.equal to the standard deviation. b.always a positive number. c.always equal to zero. d.the mean of the distribution.

d.the mean of the distribution.

A negative value of Z indicates that a.the data has a negative mean. b.a mistake has been made in computations, since Z cannot be negative. c.the number of standard deviations an observation is above the mean. d.the number of standard deviations an observation is below the mean.

d.the number of standard deviations an observation is below the mean.