Question 1

A random variable cannot be both continuous and
A) discrete.
B) uniform.
C) normal.
D) skewed.

Accepted Answer

A random variable cannot be both continuous and discrete because these are mutually exclusive categories. A continuous random variable can take any value within a range, while a discrete random variable can only take specific, distinct values.

Question 2

Which one of the following probabilities is a cumulative probability?
A) The probability that there are exactly 4 people with Type O+ blood in a sample of 10 people.
B) The probability of exactly 3 heads in 6 flips of a coin.
C) The probability that the accumulated annual rainfall in a certain city next year, rounded to the nearest inch, will be 18 inches.
D) The probability that a randomly selected woman's height is 67 inches or less.

Accepted Answer

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. Choice D, which is about the probability of a woman's height being 67 inches or less, fits this definition as it accumulates all probabilities of heights up to and including 67 inches.

Question 3

Keeping in mind that a normal distribution is a model for a continuous variable, which one of the following variables cannot possibly have a normal distribution?
A) People's opinions about a new tax law (favor or oppose)
B) Weights of 5-year-old male children
C) Handspans of adult females
D) Ounces of soda in cans labeled as having 12 ounces

Accepted Answer

Opinions about a new tax law, being categorical (favor or oppose), cannot have a normal distribution since normal distributions model continuous variables, not discrete categories.

Question 4

The time taken to answer an exam question for a randomly chosen student has a uniform probability distribution from 1 minute to 5 minutes. What is the probability that the time to answer is no more than 2 minutes?
A) 0.20
B) 0.25
C) 0.40
D) 0.75

Accepted Answer

The uniform distribution from 1 to 5 minutes means any time within this range is equally likely. The range is 5 - 1 = 4 minutes. The time "no more than 2 minutes" covers the range from 1 to 2 minutes, which is 1 minute. The probability is the desired range (1 minute) over the total range (4 minutes), so 1/4 = 0.25.

Question 5

The time taken to deliver a pizza has a uniform probability distribution from 20 minutes to 60 minutes. What is the probability that the time to deliver a pizza is at least 25 minutes?
A) 0.125
B) 0.300
C) 0.700
D) 0.875

Accepted Answer

The uniform distribution means that every outcome in the range from 20 to 60 minutes is equally likely. The range of possible delivery times is 60 - 20 = 40 minutes. If the delivery time is at least 25 minutes, we are looking at the range from 25 to 60 minutes, which is 60 - 25 = 35 minutes. The probability is the favorable range (35 minutes) over the total range (40 minutes), which is 35/40 = 0.875.

Question 6

Use the following information for questions:
 The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds.
-What is the standardized score (z-score) for a boot-up time of x =30 seconds?
A) -2.0
B) 0.0
C) 1.0
D) 2.0

Accepted Answer

The z-score is calculated using the formula z = (x - μ) / σ, where x is the value in question, μ is the mean, and σ is the standard deviation. For x = 30 seconds, μ = 30 seconds, and σ = 5 seconds, the z-score is (30 - 30) / 5 = 0 / 5 = 0.0.

Question 7

Use the following information for questions:
 The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds.
-What is the standardized score (z-score) for a boot-up time of x =20 seconds?
A) -2.0
B) 0.0
C) 1.0
D) 2.0

Accepted Answer

The z-score is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. For x = 20, μ = 30, and σ = 5, the z-score is (20 - 30) / 5 = -2.0.

Question 8

Use the following information for questions:
 The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds.
-What is the standardized score (z-score) for a boot-up of time x =35 seconds?
A) -2.0
B) 0.0
C) 1.0
D) 2.0

Accepted Answer

The z-score is calculated using the formula $z = \frac{x - \mu}{\sigma}$, where $x$ is the value in question, $\mu$ is the mean, and $\sigma$ is the standard deviation. Plugging in the given values, $z = \frac{35 - 30}{5} = 1.0$.

Question 9

Use the following information for questions:
 Find the requested probability for the standard normal random variable Z.
-What is the probability that Z is less than or equal to 2, P(Z $\leq$ 2)?
A) 0.0228
B) 0.2000
C) 0.5000
D) 0.9772

Accepted Answer

The probability that Z is less than or equal to 2 in a standard normal distribution is found by looking up the value in a Z-table or using a calculator designed for normal distributions. The value corresponds to 0.9772, indicating that there is a 97.72% chance that Z will be less than or equal to 2.

Question 10

Use the following information for questions:
 Find the requested probability for the standard normal random variable Z.
-What is the probability that Z is greater than 2, P(Z > 2)?
A) 0.0228
B) 0.2000
C) 0.5000
D) 0.9772

Accepted Answer

The probability that Z is greater than 2 in a standard normal distribution can be found using a Z-table or standard normal distribution table, which shows that P(Z > 2) is approximately 0.0228. This represents the area under the curve to the right of Z = 2.

Question 11

Use the following information for questions:
 Find the requested probability for the standard normal random variable Z.
-What is the probability that Z is between -1 and 1, P(-1 $\leq$ Z $\leq$ 1)?
A) 0.1587
B) 0.3174
C) 0.6826
D) 0.8413

Accepted Answer

The probability that Z is between -1 and 1 in a standard normal distribution is approximately 0.6826, which corresponds to about 68% of the data under the curve falling within one standard deviation from the mean.

Question 12

Use the following information for questions:
 Find the requested probability for the standard normal random variable Z.
-What is the probability that Z is between -1.2 and 1.45, P(-1.2 $\leq$Z$\leq$ 1.45)?
A) 0.0303
B) 0.7740
C) 0.8041
D) 0.8114

Accepted Answer

The answer of Use the following information for questions:
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Question 13

Assuming a standard normal distribution is appropriate, what is the approximate probability that a z-score is greater than or equal to 2.33? Said another way, what is P(Z $\geq$ 2.33)?
A) 0.99
B) 0.01
C) 0.15
D) 0.25

Accepted Answer

The area under the standard normal curve to the right of z = 2.33 is approximately 0.01, indicating that there is about a 1% chance of observing a z-score greater than or equal to 2.33.

Question 14

Suppose that vehicle speeds at an interstate location have a normal distribution with a mean equal to 70 mph and standard deviation equal to 8 mph. What is the z-score for a speed of 64 mph?
A) -6
B) -0.75
C) +0.75
D) +6

Accepted Answer

The z-score is calculated using the formula $z = \frac{(X - \mu)}{\sigma}$, where $X$ is the value in question, $\mu$ is the mean, and $\sigma$ is the standard deviation. For a speed of 64 mph, the z-score is $z = \frac{(64 - 70)}{8} = -\frac{6}{8} = -0.75$.

Question 15

Heights of college women have a distribution that can be approximated by a normal curve with a mean of 65 inches and a standard deviation equal to 3 inches. About what proportion of college women are between 65 and 67 inches tall?
A) 0.75
B) 0.50
C) 0.25
D) 0.17

Accepted Answer

Using the properties of a normal distribution, we know that approximately 68% of the data falls within one standard deviation of the mean. Since the question asks for the proportion of women between 65 and 67 inches tall, we are looking at the range from the mean to one standard deviation above the mean. This range captures about half of the 68%, which is approximately 34%. However, the closest option to 34% is 25% (option C), indicating a slight misunderstanding in the direct application of the rule. The correct interpretation should consider that the exact proportion in this specific range can be more accurately found using the standard normal distribution table or Z-scores, which would typically show that the area from the mean to one standard deviation above encompasses slightly less than 34%. However, without precise calculation tools or options that closely match the expected percentage, the best estimate given the options would be around 34%, suggesting a need for clarification in the answer choices provided.

Question 16

Verbal SAT scores have approximately a normal distribution with mean equal to 500 and standard deviation equal to 100. The 95^thpercentile of z-scores is z = 1.65. What is the 95^th percentile of verbal SAT scores? A) 335 B) 500 C) 600 D) 665

Accepted Answer

To find the 95th percentile of verbal SAT scores, we use the formula for converting z-scores to original scores: X = μ + zσ. Here, μ = 500, σ = 100, and z = 1.65. Plugging these values in gives X = 500 + (1.65)(100) = 500 + 165 = 665.

Question 17

Verbal SAT scores have a mean of 500 and a standard deviation of 100. Which of the following describes how to find the proportion of Verbal SAT scores that are greater than 600?
A) Find the area to the left of z = 1 under a standard normal curve.
B) Find the area between z =-1 and z = 1 under a standard normal curve.
C) Find the area to the right of z = 1 under a standard normal curve.
D) Find the area to the right of z = -1 under a standard normal curve.

Accepted Answer

To find the proportion of scores greater than 600, you first calculate the z-score for 600 using the formula $z = \frac{(X - \mu)}{\sigma}$, where $X$ is 600, $\mu$ is 500, and $\sigma$ is 100. This gives $z = \frac{(600 - 500)}{100} = 1$. The proportion of scores greater than 600 is then found by looking at the area to the right of $z = 1$ on a standard normal curve.

Question 18

Pulse rates of adult men are approximately normal with a mean of 70 and a standard deviation of 8. Which choice correctly describes how to find the proportion of men that have a pulse rate greater than 78?
A) Find the area to the left of z = 1 under a standard normal curve.
B) Find the area between z = -1 and z = 1 under a standard normal curve.
C) Find the area to the right of z =1 under a standard normal curve.
D) Find the area to the right of z = - under a standard normal curve.

Accepted Answer

To find the proportion of men with a pulse rate greater than 78, we first calculate the z-score for 78: $z = \frac{78 - 70}{8} = 1$. The proportion of men with a pulse rate greater than 78 is found by looking at the area to the right of z = 1 under a standard normal curve.

Question 19

Scores on an achievement test had an average of 70 and a standard deviation of 10. Serena's score was 85. Assuming the scores have approximately a normal distribution, about what proportion of students scored lower than Serena?
A) 0.93
B) 0.07
C) 0.84
D) 0.68

Accepted Answer

Serena's score is 1.5 standard deviations above the mean (85 - 70 = 15, 15/10 = 1.5). In a normal distribution, about 93% of the data falls below a score that is 1.5 standard deviations above the mean.

Question 20

Weights of females have approximately a normal distribution with mean 135 lbs. and standard deviation 20 lbs. Allison weighs 145 lbs. What is the z-score for her weight?
A) 10
B) 1.50
C) 0.50
D) 0.20

Accepted Answer

The z-score is calculated by subtracting the mean from the individual score and then dividing by the standard deviation: $z = \frac{(X - \mu)}{\sigma}$. For Allison, $z = \frac{(145 - 135)}{20} = \frac{10}{20} = 0.50$.

Quiz 8: Random Variables

A random variable cannot be both continuous and

Which one of the following probabilities is a cumulative probability?

Keeping in mind that a normal distribution is a model for a continuous variable, which one of the following variables cannot possibly have a normal distribution?

The time taken to answer an exam question for a randomly chosen student has a uniform probability distribution from 1 minute to 5 minutes. What is the probability that the time to answer is no more than 2 minutes?

The time taken to deliver a pizza has a uniform probability distribution from 20 minutes to 60 minutes. What is the probability that the time to deliver a pizza is at least 25 minutes?

Use the following information for questions: The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds. -What is the standardized score (z-score) for a boot-up time of x =30 seconds?

Use the following information for questions: The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds. -What is the standardized score (z-score) for a boot-up time of x =20 seconds?

Use the following information for questions: The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds. -What is the standardized score (z-score) for a boot-up of time x =35 seconds?

Use the following information for questions: Find the requested probability for the standard normal random variable Z. -What is the probability that Z is less than or equal to 2, P(Z $\leq$ 2) ?

Use the following information for questions: Find the requested probability for the standard normal random variable Z. -What is the probability that Z is greater than 2, P(Z > 2) ?

Use the following information for questions: Find the requested probability for the standard normal random variable Z. -What is the probability that Z is between -1 and 1, P(-1 $\leq$ Z $\leq$ 1) ?

Use the following information for questions: Find the requested probability for the standard normal random variable Z. -What is the probability that Z is between -1.2 and 1.45, P(-1.2 $\leq$ Z $\leq$ 1.45) ?

Assuming a standard normal distribution is appropriate, what is the approximate probability that a z-score is greater than or equal to 2.33? Said another way, what is P(Z $\geq$ 2.33) ?

Suppose that vehicle speeds at an interstate location have a normal distribution with a mean equal to 70 mph and standard deviation equal to 8 mph. What is the z-score for a speed of 64 mph?

Heights of college women have a distribution that can be approximated by a normal curve with a mean of 65 inches and a standard deviation equal to 3 inches. About what proportion of college women are between 65 and 67 inches tall?

Verbal SAT scores have approximately a normal distribution with mean equal to 500 and standard deviation equal to 100. The 95^thpercentile of z-scores is z = 1.65. What is the 95^th percentile of verbal SAT scores?

Verbal SAT scores have a mean of 500 and a standard deviation of 100. Which of the following describes how to find the proportion of Verbal SAT scores that are greater than 600?

Pulse rates of adult men are approximately normal with a mean of 70 and a standard deviation of 8. Which choice correctly describes how to find the proportion of men that have a pulse rate greater than 78?

Scores on an achievement test had an average of 70 and a standard deviation of 10. Serena's score was 85. Assuming the scores have approximately a normal distribution, about what proportion of students scored lower than Serena?

Weights of females have approximately a normal distribution with mean 135 lbs. and standard deviation 20 lbs. Allison weighs 145 lbs. What is the z-score for her weight?