Chapter 6. ECON 2030 Stats

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?

. 0.4866

For a uniform probability density function,

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

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?

.0049 - A z-score of 2.58 corresponds to 30 successes. P(z > 2.58) = 0.0049

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 90 or more say that they watch TV in the evening?

.006 -he mean of this distribution is 80 and the standard deviation is 4, therefore, 90 successes correspond to a z-score of 2.5. P(z ≥ 2.5) = 0.006

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?

.2397 - 1-e - 5/3.5= .7603 so p (X>5)= 1- .7603 = .2397

The height of the probability density function for a uniform distribution ranging between 2 and 6 is

.25 -The total area must equal one. The base is 4, therefore the height must be 1/4 or 0.25. See Section 6.1, Uniform 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?

.49 -The mean of this distribution is 80 and the standard deviation is 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

For the standard normal probability distribution, the area to the left of the mean is

.5

The random variable X is known to be uniformly distributed between 2 and 12. Compute P(X ≥ 6).

.6 -- P(X ≥ 6) = (base)(height) = (6)(1/10)

The random variable X is known to be uniformly distributed between 2 and 12. Compute P(X > 6).

.6 --- P(X > 6) = (base)(height) = (6)(1/10) = 0.6. The probability of a continuous random variable assuming a value in any interval is the same whether or not the endpoints are included.

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?

.6% -The z-score that corresponds with the car getting 100 mpg is Z = 2.5. P(Z ≥ 2.5) = 0.06%

Z is a standard normal random variable. Compute P(-1.5 < Z < 1.5).

.87 .9332- .0668

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?

.89 - The mean of this distribution is 80 and the standard deviation is 4, therefore, 85 successes correspond to a z-score of 1.25. P(z < 1.25) = 0.89.

Z is a standard normal random variable. Compute P(Z > -1.75).

.96 1 - .0401 = .96

The random variable X is known to be uniformly distributed between 2 and 12. Compute P(X = 6).

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.

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?

0.1283

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 standard deviation of this distribution?

1.5 mins

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?

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%

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.

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?

12.25 -A property of the exponential distribution is that the mean of the distribution and the standard deviation of the distribution are equal, so the standard deviation is 3.5. The variance is the square of the standard deviation, so the variance = 3.52 = 12.25

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?

2.25 - 1.52^2 = 2.25

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. What percent of the time does he exceed 13,000 steps?

2.28%

The random variable X is known to be uniformly distributed between 2 and 12. Compute the standard deviation of X.

2.2887 -The variance of a random variable that is uniformly distributed is Var(X) = (b - a)2/12 = (12 - 2)2 = 12 = 8.333.

A binomial probability distribution has p = 0.25 and n = 80. What is the mean of this distribution?

20 - The mean of a binomial distribution = np = (80)(0.25) = 20

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?

3 - 4.78% ( 100-4.78 = 95.22 [percentile] is 1.66 then plug in the x-u/ deviation

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 standard deviation of this distribution?

3.5 mins

A binomial probability distribution has p = 0.25 and n = 80. What is the standard deviation of this distribution?

3.87 - square root (80)(.25)(.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?

4.6% - Use the continuity correction factor to calculate this probability. Find P(84.5 ≤ X ≤ 85.5). This equates to P(1.13 ≤ z ≤ 1.38) = 10.3%

The random variable X is known to be uniformly distributed between 2 and 12. Compute E(X).

7 1/10 [12^2 - 2^2] (1/10)^2 1/20 1/20 [ 144-4] = 140/20 7

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?

75 -he 50th percentile of a normal distribution is located at its mean.

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?

90% - A z-score of 1.29 corresponds to 25 successes. P(z ≤ 1.29) = 0.90

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?

99.7% -The z-score for 15,000 steps is z = 3.33. P(z ≤ 3.33) = 99.7%

A normal distribution with a mean of 0 and a standard deviation of 1 is called

a standard normal distribution.

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

b. exponential probability 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

continuity correction factor. - 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 continuity correction factor.

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

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

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

d. an exponential probability distribution.

A continuous probability distribution that is useful in describing the time, or space, between occurrences of an event is a(n)

exponential probability distribution

When np ≥ 5 and n(1 − p) ≥ 5, the binomial probabilities can be approximated by a

normal distribution.

The mean of a binomial distribution can be calculated using the formula

np

The standard deviation of a binomial distribution can be calculated using the formula

square root (np (1-p)

The center of a normal curve is

the mean of the distribution.

A negative value of Z indicates that

the number of standard deviations an observation is below the mean

In a standard normal distribution, what Z-score corresponds to the 75th percentile?

z= .67

Which of the following is not a characteristic of the normal probability distribution?

The standard deviation must be 1.