Question 1

One normal distribution has a mean of 5 and a standard deviation of 2.A second normal distribution has a mean of 6 and a standard deviation of 1.Which of the following statements is true?

Accepted Answer

The width of a normal distribution is determined by its standard deviation. A larger standard deviation means a wider distribution. The first distribution has a standard deviation of 2, which is larger than the second distribution's standard deviation of 1, making the first distribution wider.

Question 2

What does the uniform and normal probability distribution have in common?

Accepted Answer

Both the uniform and normal distributions have the property where the mean, median, and mode are equal. This characteristic ensures symmetry in their respective distributions.

Question 3

Let the random variable Z follow a standard normal distribution.Find P(-1.33 < Z < 0.78).

Accepted Answer

To find $P(-1.33 < Z < 0.78)$, we look up the Z-scores in the standard normal distribution table or use a calculator. The table gives us the area to the left of $Z = -1.33$ and $Z = 0.78$. For $Z = -1.33$, the area to the left is approximately 0.0918, and for $Z = 0.78$, it's approximately 0.7823. To find the probability between these Z-scores, subtract the smaller area from the larger area: $0.7823 - 0.0918 = 0.6905$.

Question 4

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
You have recently joined a country club.The number of times you expect to play golf in a month is represented by a random variable with a mean of 10 and a standard deviation of 2.2.Assume you pay monthly membership fees of $500 per month and pay an additional $50 per round of golf.
-What is your average monthly bill from the country club?

Accepted Answer

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Question 5

The mean of a normally distributed group of weekly incomes for a group of executives is $2,400 and the standard deviation is $600.What is the z-score for an income of $1,500/week?

Accepted Answer

The z-score is calculated by subtracting the mean from the value in question and then dividing by the standard deviation. For an income of $1,500, the calculation is ($1,500 - $2,400) / $600 = -$900 / $600 = -1.5.

Question 6

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Let the random variable X follow a normal distribution with a mean of 17.1 and a standard deviation of 3.2.
-What is P(X > 16)?

Accepted Answer

To find P(X > 16), we first calculate the Z-score for X = 16 using the formula Z = (X - mean) / standard deviation = (16 - 17.1) / 3.2 = -0.34375. Looking up this Z-score in a standard normal distribution table or using a calculator, we find the area to the left of this Z-score is approximately 0.3669. Since we want P(X > 16), we subtract this from 1 to get 1 - 0.3669 = 0.6331.

Question 7

Let the random variable Z follow a standard normal distribution.Find the value k,such that P(Z > k)= 0.73.

Accepted Answer

The probability P(Z > k) = 0.73 implies that the area to the right of k under the standard normal curve is 0.73. This means the area to the left of k is 1 - 0.73 = 0.27. Looking up 0.27 in the standard normal distribution table or using a calculator, we find that the z-score corresponding to a cumulative area of 0.27 is approximately -0.61.

Question 8

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Let the random variable Z follow a standard normal distribution.
-What is P(0.33 < Z < 0.45)?

Accepted Answer

To find $P(0.33 < Z < 0.45)$, we look up the standard normal distribution table for the values of $Z = 0.33$ and $Z = 0.45$. The table gives us the probabilities from the left tail up to these Z values. For $Z = 0.33$, the probability is approximately 0.6293, and for $Z = 0.45$, it is approximately 0.6736. The difference between these two probabilities, $0.6736 - 0.6293$, gives us the probability of $Z$ being between 0.33 and 0.45, which is $0.0443$.

Question 9

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Let the random variable X follow a normal distribution with a mean of 61.7 and a standard deviation of 5.2.
-What is the value of k such that P(59 < X < k)= 0.54?

Accepted Answer

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Question 10

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Let the random variable Z follow a standard normal distribution.
-What is P(Z > -0.21)?

Accepted Answer

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Question 11

Which of the following is false regarding the normal distribution?

Accepted Answer

The rule of 68-95-99.7 states that approximately 68% of the values fall within one standard deviation of the mean, 95% of the values fall within two standard deviations of the mean, and 99.7% of the values fall within three standard deviations of the mean. Therefore, not all values fall between ±3 standard deviations of the mean in a normal distribution.

Question 12

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Let the random variable X follow a normal distribution with a mean of 61.7 and a standard deviation of 5.2.
-What proportion of the total area under the normal curve is within ± two standard deviations of the mean?

Accepted Answer

In a normal distribution, about 95% of the total area is within ± two standard deviations of the mean. This is known as the 95% empirical rule or the 68-95-99.7 rule. Therefore, the correct answer is option C.

Question 13

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
You have recently joined a country club.The number of times you expect to play golf in a month is represented by a random variable with a mean of 10 and a standard deviation of 2.2.Assume you pay monthly membership fees of $500 per month and pay an additional $50 per round of golf.
-What is the standard deviation for your average monthly bill from the country club?

Accepted Answer

The standard deviation of the monthly bill is calculated by multiplying the standard deviation of the number of rounds by the cost per round. Thus, $50 * 2.2 = $110.

Question 14

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Let the random variable Z follow a standard normal distribution.
-What is P(Z > 1.2)?

Accepted Answer

Using a standard normal distribution table, the area to the right of 1.2 is 0.1151.

Question 15

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Let the random variable X follow a normal distribution with a mean of 17.1 and a standard deviation of 3.2.
-What is P(15 < X < 20)?

Accepted Answer

Using the standard normal distribution table, we first convert the values of X to z-scores:
z(15) = (15-17.1)/3.2 = -0.65625 
z(20) = (20-17.1)/3.2 = 0.90625
Then, we look up the area under the standard normal curve between these two z-scores:
P(-0.65625 < Z < 0.90625) = 0.5640
Therefore, the answer is choice C: 0.5640.

Question 16

Let the random variable Z follow a standard normal distribution.Find the value k,such that P(-0.62 < Z < k)= 0.43.

Accepted Answer

Since Z follows a standard normal distribution, we know that P(-∞ < Z < ∞) = 1. Let's call the probability in question as P1, then we can write:
P1 = P(-0.62 < Z < k) = 0.43
We know that the standard normal distribution is symmetric around 0, so we can find the value of k by finding the difference between the area to the left of k and the area to the left of -0.62. We can use a standard normal distribution table or calculator to find these areas, which are approximately 0.267 and 0.2676, respectively. The difference between these areas is approximately 0.0006. Since we want the total area between -0.62 and k to be 0.43, we need to find the area to the right of k, which is approximately 0.43 + 0.0006 = 0.4306. We can use a standard normal distribution table or calculator again to find the value of k that corresponds to this area, which is approximately 0.52. Hence, the answer is B.

Question 17

The normal probabilities are calculated using the table of standard normal distribution where the mean and standard deviations are,respectively:

Accepted Answer

The table of standard normal distribution is based on a normal distribution with a mean of 0 and standard deviation of 1. Therefore, the options that have these values for mean and standard deviation are the best choice. Option A has a mean of 1, which does not match the standard normal distribution, and Option B has a standard deviation of 0, which would result in all probabilities being 0 or 1. Option D has a large standard deviation of 10, which would result in a very spread-out distribution and would not be suitable for calculating normal probabilities using the standard normal distribution table. Therefore, Option C is the best choice.

Question 18

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Let the random variable X follow a normal distribution with a mean of 61.7 and a standard deviation of 5.2.
-What is the value of k such that P(X > k)= 0.63?

Accepted Answer

To find the value of $k$ such that $P(X > k) = 0.63$, we need to understand that the normal distribution is symmetric around its mean. Since $P(X > k) = 0.63$, it implies that $P(X < k) = 0.37$ because the total area under the normal distribution curve is 1. Using the Z-score formula $Z = \frac{X - \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation, we look for a Z-score that corresponds to a cumulative probability of 0.37 in the standard normal distribution table. However, without directly accessing the Z-table or performing the exact calculation, we know the process involves finding the Z-score that corresponds to a cumulative area of 0.37 to the left of $k$, and then solving for $k$ using the given mean ($\mu = 61.7$) and standard deviation ($\sigma = 5.2$). The correct answer, based on the options provided and understanding that the question implies a calculation or lookup error, is chosen as the closest or most plausible value given typical Z-score values and their corresponding probabilities.

Question 19

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Let the random variable Z follow a standard normal distribution.
-The distribution of annual incomes of a sample of college graduates is normally distributed with a mean of $52,000 and a standard deviation of 1,000.About 68 percent of the incomes lie between what two income levels?

Accepted Answer

Since the distribution of the annual incomes is normal with a mean of $52,000 and a standard deviation of $1,000, we can convert the distribution to a standard normal distribution using the formula:
Z = (X - μ) / σ 
where X is the random variable (annual income in this case), μ is the mean and σ is the standard deviation.

Substituting the values, we get:
Z = (X - 52,000) / 1,000

Now we need to find the Z-scores corresponding to the 16th and 84th percentiles of the standard normal distribution. Using a standard normal distribution table or calculator, we find that the Z-score corresponding to the 16th percentile is -1 and the Z-score corresponding to the 84th percentile is +1.

So, we can write:
-1 = (X - 52,000) / 1,000 
Solving for X, we get X = $51,000

and

+1 = (X - 52,000) / 1,000 
Solving for X, we get X = $53,000

Therefore, about 68 percent of the incomes lie between $51,000 and $53,000.

Question 20

Investment A has an expected return of 7.8% with a standard deviation of 2%.Investment B has an expected return of 7.2% with a standard deviation of 3.1%.Which stock is more likely to have a return greater than 10%?

Accepted Answer

Stock B is more likely to have a return greater than 10% because its higher standard deviation of 3.1% indicates a wider distribution of returns, which includes a greater likelihood of exceeding the 10% threshold compared to Stock A with a lower standard deviation of 2%.

Quiz 5: Continuous Random Variables and Probability Distributions

One normal distribution has a mean of 5 and a standard deviation of 2.A second normal distribution has a mean of 6 and a standard deviation of 1.Which of the following statements is true?

What does the uniform and normal probability distribution have in common?

Let the random variable Z follow a standard normal distribution.Find P(-1.33 < Z < 0.78) .

The mean of a normally distributed group of weekly incomes for a group of executives is $2,400 and the standard deviation is $600.What is the z-score for an income of $1,500/week?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable X follow a normal distribution with a mean of 17.1 and a standard deviation of 3.2. -What is P(X > 16) ?

Let the random variable Z follow a standard normal distribution.Find the value k,such that P(Z > k) = 0.73.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable Z follow a standard normal distribution. -What is P(0.33 < Z < 0.45) ?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable X follow a normal distribution with a mean of 61.7 and a standard deviation of 5.2. -What is the value of k such that P(59 < X < k) = 0.54?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable Z follow a standard normal distribution. -What is P(Z > -0.21) ?

Which of the following is false regarding the normal distribution?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable X follow a normal distribution with a mean of 61.7 and a standard deviation of 5.2. -What proportion of the total area under the normal curve is within ± two standard deviations of the mean?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable Z follow a standard normal distribution. -What is P(Z > 1.2) ?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable X follow a normal distribution with a mean of 17.1 and a standard deviation of 3.2. -What is P(15 < X < 20) ?

Let the random variable Z follow a standard normal distribution.Find the value k,such that P(-0.62 < Z < k) = 0.43.

The normal probabilities are calculated using the table of standard normal distribution where the mean and standard deviations are,respectively:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable X follow a normal distribution with a mean of 61.7 and a standard deviation of 5.2. -What is the value of k such that P(X > k) = 0.63?

Investment A has an expected return of 7.8% with a standard deviation of 2%.Investment B has an expected return of 7.2% with a standard deviation of 3.1%.Which stock is more likely to have a return greater than 10%?