Question 1

If the random variable of x is normally distributed,____% of all possible observed values of x will be within three standard deviations of the mean.
A)68.26
B)95.44
C)99.73
D)100
E)Cannot be determined without knowing the actual mean and standard deviation.

Accepted Answer

According to the empirical rule, 99.73% of all possible values fall within three standard deviations of the mean for a normally distributed variable. This holds true regardless of the actual mean and standard deviation.

Question 2

Which of the following statements is not a property of the normal probability distribution?
A)The normal distribution is symmetric.
B)95.44% of all possible observed values of the random variable x are within three standard deviations of the population mean.
C)The mean,median,and mode are equal.
D)The area under the normal curve to the right of the mean is equal to the area under the normal curve to the left of the mean.
E)68.28% of all possible observed values of the random variable x are within one standard deviation of the population mean.

Accepted Answer

95.44% of all possible observed values of the random variable x are within two standard deviations of the population mean, not three.

Question 3

The normal approximation of the binomial distribution is appropriate when:
A)np ≥ 5
B)nq ≥ 5
C)np ≤ 5
D)nq ≤ 5 and np ≤ 5
E)np ≥ 5 and nq ≥ 5

Accepted Answer

The normal approximation of the binomial distribution is appropriate when both np and nq are greater than or equal to 5 (E). This condition ensures that there are enough successes and failures in the sample to approximate a normal distribution.

Question 4

A study shows that employees that begin their work day at 9:00 a.m.vary their times of arrival uniformly from 8:40 a.m.to 9:30 a.m.The probability that a randomly chosen employee reports to work between 9:00 and 9:10 is:
A)0.4
B)0.2
C)0.1
D)0.3
E)0.167

Accepted Answer

The total time interval from 8:40 a.m. to 9:30 a.m. is 50 minutes. The interval from 9:00 to 9:10 a.m. is 10 minutes. Since the arrival times are uniformly distributed, the probability is the ratio of the desired interval to the total interval, which is 10/50 = 0.2.

Question 5

Which one of the following variables could be modeled using a continuous probability distribution?
A)The number of students in a university course.
B)The number of coffees you buy during a school year.
C)The amount of time you wait in line to buy a coffee.
D)The number of time somebody sneezes during a 1 hour class.
E)The number of times somebody shaves during one week.

Accepted Answer

The amount of time one waits in line can take on any value along a continuous range, and can therefore be modeled using a continuous probability distribution. The other variables are discrete in nature and would be better modeled using a discrete probability distribution.

Question 6

The relationship between the standard normal random variable z and normal random variable X is that:
A)Only the normal random variable X is continuous.
B)Only the standard normal variable z is continuous.
C)The standard normal variable z counts the number of standard deviations that the value of the normal random variable X is away from its mean.
D)The values of the standard normal random variable z cannot be negative.
E)The values of the normal random variable X cannot be negative.

Accepted Answer

The standard normal variable z is calculated using the formula (X-μ)/σ, where X is a normal random variable with mean μ and standard deviation σ. Therefore, the standard normal variable z measures the number of standard deviations that X is away from its mean. Both the standard normal variable z and normal random variable X are continuous. The values of both z and X can be either positive or negative.

Question 7

A student's grade on an examination was transformed to a z value which is negative.Therefore,we know that the student scored:
A)Higher than 16% of the class
B)Higher than 45% of the class
C)Above the first quartile
D)Lower than 16% of the class
E)Below the mean

Accepted Answer

A negative z-score indicates that the student's score was below the mean of the class. Z-scores measure how many standard deviations an element is from the mean. A negative value means the score is less than the mean.

Question 8

The area under the normal curve between z = 0 and z = 1 is ________________ the area under the normal curve between z = 1 and z = 2.
A)less than
B)greater than
C)equal to
D)always equal to
E)sometimes equal to

Accepted Answer

The area under the normal curve between z = 0 and z = 1 is greater than the area between z = 1 and z = 2 because the normal distribution is denser closer to the mean (z = 0), and the probability density decreases as one moves away from the mean.

Question 9

A standard normal distribution has a mean of _____ and standard deviation of _____.
A)zero;zero
B)zero;one
C)one;one
D)one;zero
E)zero;three

Accepted Answer

A standard normal distribution has a mean of zero and a standard deviation of one. This means that the distribution has a symmetric bell-shaped curve centered at zero, and the spread of the data is defined by the standard deviation of one. None of the other choices accurately describe a standard normal distribution.

Question 10

A property of continuous probability distributions is that:
A)Like discrete random variables,the probability distribution can be approximated by a smooth curve.
B)Probabilities for continuous variables can be approximated using discrete random variables.
C)Unlike discrete random variables,probabilities can be found using tables.
D)Unlike discrete random variables,the probability that a continuous random variable equals a specific value is zero;that is,P(X = x)= 0.
E)Unlike discrete random variables,probabilities for continuous random variables can sum to a value greater than 1.

Accepted Answer

This is the property of continuous probability distributions that distinguishes them from discrete random variables. Since a continuous random variable can take on an infinite number of values within an interval, the probability that it equals a specific value is zero. This is because the area under the probability density function between two neighboring points on the interval is what gives the probability of the variable being in that range, and the width of a single point on the interval is zero.

Question 11

The fill weight of a certain brand of adult cereal is normally distributed with a mean of 910 grams and a standard deviation of 5 grams.
-We calculated the value of z for a specific box of this brand of cereal and the z value was negative.This negative z value indicates that:
A)We made a mistake in our calculations,as z must always be a nonnegative number.
B)The fill weight exceeds 910 grams.
C)The fill weight is less than 910 grams
D)The fill weight is equal to 910 grams
E)The fill weight may be more or less than 910 grams depending on the value of the standard deviation.

Accepted Answer

A negative z value indicates that the observed fill weight is less than the mean fill weight of 910 grams. As z is calculated using the formula (observed value - mean) / standard deviation, a negative value indicates that the observed value is less than the mean. Therefore, option C is the correct answer. Option A is incorrect as z can be negative if the observed value is less than the mean. Option B is incorrect as it contradicts the negative z value obtained. Option D is also incorrect as it assumes that the z value is zero, which is not the case. Option E is also incorrect as the sign of z does not depend on the standard deviation.

Question 12

Let X be a binomial random variable with n = 150 and p = 0.3.Approximate P(X $\geq$
50))
A)0.7881
B)0.9909
C)0.2119
D)0.0091
E)0.5644

Accepted Answer

The answer of Let X be a binomial random variable...

Question 13

The fill weight of a certain brand of adult cereal is normally distributed with a mean of 910 grams and a standard deviation of 5 grams.
-If we select one box of cereal at random from this population,what is the probability that it will weigh less than 900 grams?
A)4772
B)9772
C)9544
D)0456
E)0228

Accepted Answer

To find the probability that a box of cereal weighs less than 900 grams, we use the Z-score formula: Z = (X - μ) / σ, where X is the value in question (900 grams), μ is the mean (910 grams), and σ is the standard deviation (5 grams). Plugging in the values, we get Z = (900 - 910) / 5 = -10 / 5 = -2. The Z-score of -2 corresponds to a probability of 0.0228 (or 2.28%) in the standard normal distribution table.

Question 14

The fuel efficiency rating for a mid-size car is normally distributed with a mean of 32 and a standard deviation of 0.8. What is the probability that the rating for a selected mid-size car would be:
-More than 33.2?
A)0.4332
B)0.0668
C)0.9332
D)0.8664
E)0.1336

Accepted Answer

To find the probability that the rating is more than 33.2, we first calculate the Z-score: Z = (X - μ) / σ = (33.2 - 32) / 0.8 = 1.5. Using a standard normal distribution table, the probability of Z being less than 1.5 is approximately 0.9332. Therefore, the probability of being more than 33.2 is 1 - 0.9332 = 0.0668.

Question 15

If the wages of workers for a given company are normally distributed with a mean of $15 per hour,then the proportion of the workers earning more than $13 per hour:
A)Is greater than the proportion earning less than $13 per hour.
B)Is greater than the proportion earning less than $18 per hour.
C)Is less than 50%.
D)Is less than the proportion earning more than the mean wage.
E)Is less than the proportion earning more than $14 per hour.

Accepted Answer

Since the wages are normally distributed with a mean of $15 per hour, the distribution is symmetric about the mean. Therefore, the proportion of workers earning more than $13 per hour is equal to the proportion earning less than $17 per hour (which is the same distance from the mean in the opposite direction). However, $13 is closer to the mean than $17, so more workers will be earning more than $13 per hour than less than $17 per hour. Therefore, choice A is correct.

Question 16

The fuel efficiency rating for a mid-size car is normally distributed with a mean of 32 and a standard deviation of 0.8. What is the probability that the rating for a selected mid-size car would be:
-Less than 33.2?
A)0.4332
B)0.0668
C)0.9332
D)0.8664
E)0.1336

Accepted Answer

To find the probability that the rating is less than 33.2, we calculate the Z-score: $Z = \frac{33.2 - 32}{0.8} = 1.5$. Looking up 1.5 in the Z-table gives us a probability of 0.9332.

Question 17

The _____________ distribution would most likely be used to describe the distribution of time between arrivals of customers at the grocery store.
A)normal
B)exponential
C)Poisson
D)binomial
E)uniform

Accepted Answer

The exponential distribution is used to model the time between arrivals in a Poisson process. In this case, customers arriving at a grocery store can be modeled as a Poisson process, where the arrival rate is constant over time, and the time between arrivals follows an exponential distribution. Therefore, the exponential distribution would be the most appropriate choice for modeling the time between arrivals of customers at the grocery store.

Question 18

The price-to-earnings ratio for firms in a given industry is distributed according to normal distribution.In this industry,a firm with a standard normal variable value of z = 1:
A)Has an above average price-to-earning ratio.
B)Has a below average price-to-earnings ratio.
C)Has an average price-to-earnings ratio.
D)Has the minimum possible price-to-earnings ratio.
E)Has the maximum possible price-to-earnings ratio.

Accepted Answer

A standard normal variable value of z=1 corresponds to being one standard deviation above the mean in a normal distribution. Since the mean is considered as an average, being one standard deviation above the mean would correspond to being above average in this industry. Therefore, a firm with z=1 would have an above average price-to-earnings ratio.

Question 19

The fill weight of a certain brand of adult cereal is normally distributed with a mean of 910 grams and a standard deviation of 5 grams.
-If we select one box of cereal at random from this population,what is the probability that it will weigh more than 904 grams?
A)8849
B)3849
C)1151
D)7698
E)2302

Accepted Answer

The probability that a randomly selected box of cereal will weigh more than 904 grams is 0.8849, or 88.49%. This can be calculated using the normal distribution formula:P(X > 904) = 1 - P(X <= 904) = 1 - 0.8849 = 0.1151Therefore, the probability that a randomly selected box of cereal will weigh more than 904 grams is 0.1151, or 11.51%.

Question 20

If the random variable X has a mean of $\mu$
And a standard deviation $\sigma$
,then $( X - \mu ) / \sigma$
Has a mean of ____ and standard deviation of _____.
A) $\mu$ and $\sigma$
B) $\bar { X }$ and s
C)1 and 0
D)0 and 1
E)Cannot be determined without more information.

Accepted Answer

The transformation $(X - \mu) / \sigma$ standardizes the random variable X, resulting in a new variable with a mean of 0 and a standard deviation of 1. This process is known as standardization or normalization.

Quiz 5: Continuous Random Variables

If the random variable of x is normally distributed,____% of all possible observed values of x will be within three standard deviations of the mean.

Which of the following statements is not a property of the normal probability distribution?

The normal approximation of the binomial distribution is appropriate when:

A study shows that employees that begin their work day at 9:00 a.m.vary their times of arrival uniformly from 8:40 a.m.to 9:30 a.m.The probability that a randomly chosen employee reports to work between 9:00 and 9:10 is:

Which one of the following variables could be modeled using a continuous probability distribution?

The relationship between the standard normal random variable z and normal random variable X is that:

A student's grade on an examination was transformed to a z value which is negative.Therefore,we know that the student scored:

The area under the normal curve between z = 0 and z = 1 is ________________ the area under the normal curve between z = 1 and z = 2.

A standard normal distribution has a mean of _ and standard deviation of _.

A property of continuous probability distributions is that:

The fill weight of a certain brand of adult cereal is normally distributed with a mean of 910 grams and a standard deviation of 5 grams. -We calculated the value of z for a specific box of this brand of cereal and the z value was negative.This negative z value indicates that:

Let X be a binomial random variable with n = 150 and p = 0.3.Approximate P(X $\geq$ 50) )

The fill weight of a certain brand of adult cereal is normally distributed with a mean of 910 grams and a standard deviation of 5 grams. -If we select one box of cereal at random from this population,what is the probability that it will weigh less than 900 grams?

The fuel efficiency rating for a mid-size car is normally distributed with a mean of 32 and a standard deviation of 0.8. What is the probability that the rating for a selected mid-size car would be: -More than 33.2?

If the wages of workers for a given company are normally distributed with a mean of $15 per hour,then the proportion of the workers earning more than $13 per hour:

The fuel efficiency rating for a mid-size car is normally distributed with a mean of 32 and a standard deviation of 0.8. What is the probability that the rating for a selected mid-size car would be: -Less than 33.2?

The _____________ distribution would most likely be used to describe the distribution of time between arrivals of customers at the grocery store.

The price-to-earnings ratio for firms in a given industry is distributed according to normal distribution.In this industry,a firm with a standard normal variable value of z = 1:

The fill weight of a certain brand of adult cereal is normally distributed with a mean of 910 grams and a standard deviation of 5 grams. -If we select one box of cereal at random from this population,what is the probability that it will weigh more than 904 grams?

If the random variable X has a mean of $\mu$ And a standard deviation $\sigma$ ,then $( X - \mu ) / \sigma$ Has a mean of and standard deviation of _.