Question 1

The total area under the normal probability distribution is equal to which of the following?

Accepted Answer

The total area under the normal probability distribution curve is equal to 1.0, which represents the entire probability of all outcomes in a normal distribution.

Question 2

For the equation for a normal distribution shown, what are the mean and variance?
?
$f(x)=\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-16)^{2}}{32}}$

Accepted Answer

The mean ($\mu$) is the value in the term $(x - \mu)$, which is 16, and the variance ($\sigma^2$) is twice the denominator in the exponent, which is 32, making the variance 16.

Question 3

For the equation for a normal distribution shown, what are the mean and standard deviation?
?
$f(x)=\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-40)^{2}}{32}}$

Accepted Answer

The equation is in the form of a normal distribution: $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$. Here, $\mu = 40$ and $2\sigma^2 = 32$, so $\sigma^2 = 16$. Thus, the mean is 40 and the variance is 16.

Question 4

Which equation shown is correct for a normal distribution with a mean of 20 and a variance of 10?

Accepted Answer

The correct equation for a normal distribution is given by $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$, where $\mu$ is the mean and $\sigma^2$ is the variance. For a mean ($\mu$) of 20 and a variance ($\sigma^2$) of 10, the correct substitution results in $f(x) = \frac{1}{\sqrt{20\pi}} e^{-\frac{(x-20)^2}{20}}$, which after adjusting the denominator inside the exponent to match the format of the given options, becomes $f(x) = \frac{1}{\sqrt{20\pi}} e^{-\frac{(x-20)^2}{200}}$, matching option B.

Question 5

Which equation shown is correct for a normal distribution with a median of 25 and a standard deviation of 5?

Accepted Answer

The correct equation for a normal distribution with mean (median) 25 and standard deviation 5 is $f(x)=\frac{1}{\sqrt{50 \pi}} e^{\frac{-(x-25)^{2}}{50}}$, where the variance is $5^2 = 25$.

Question 6

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

Accepted Answer

The normal distribution is a continuous distribution, not a discrete one. This means it takes on values within a continuous range rather than discrete, separate values. Choices A, C, and D are true properties of the normal distribution: it is continuous, symmetric, and its mean, median, and mode are all equal.

Question 7

Which of the following is not a good rule of thumb for data values exhibiting a normal distribution?

Accepted Answer

For a normal distribution, exactly half of the values lie above the mean due to its symmetric nature, but not exactly half of the values lie within one standard deviation of the mean. Instead, about 68% of the data falls within one standard deviation of the mean.

Question 8

For values from a data set, when we take the value, subtract the mean, and then divide by the standard deviation, we create new values called which of the following??

Accepted Answer

This process is known as standardization, and the new values created are called "Standard normal deviates." This transforms the data into a standard normal distribution with a mean of 0 and a standard deviation of 1.

Question 9

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are less than 93?

Accepted Answer

The answer of Consider a population of 900 values that...

Question 10

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are greater than 84?

Accepted Answer

The z-score for a value of 84 is (84 - 90) / 6 = -1. The area under the normal curve to the right of z = -1 is 0.8413. Therefore, 0.8413 * 900 = 757 values are greater than 84.

Question 11

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are between 87 and 99?

Accepted Answer

To find the number of values between 87 and 99 in a normally distributed population, we use the Z-score formula: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. For X = 87, Z = (87 - 90) / 6 = -0.5. For X = 99, Z = (99 - 90) / 6 = 1.5. Looking these Z-scores up in a standard normal distribution table, the area between Z = -0.5 and Z = 1.5 is approximately 0.624. Multiplying this by the total population (0.624 * 900) gives approximately 562 values.

Question 12

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value of Z below most closely corresponds to the 65th percentile?

Accepted Answer

The answer of Consider a population of 900 values that...

Question 13

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value below is closest to the value larger than exactly 55% of the population?

Accepted Answer

The answer of Consider a population of 900 values that...

Question 14

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What proportion of values are larger than 78?

Accepted Answer

To find the proportion of values larger than 78, we first calculate the z-score for 78 using the formula $z = \frac{(X - \mu)}{\sigma}$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Plugging in the given values, we get $z = \frac{(78 - 90)}{6} = -2$. Looking up a z-score of -2 in the standard normal distribution table or using a calculator, we find that the proportion of values less than 78 is approximately 0.0228. Therefore, the proportion of values larger than 78 is $1 - 0.0228 = 0.9772$.

Question 15

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. Which value below is closest to the IQR?

Accepted Answer

The answer of Consider a population of 900 values that...

Question 16

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are less than 62?

Accepted Answer

To find the number of values less than 62, we calculate the z-score for 62. The z-score is (62 - 60) / 8 = 0.25. Looking up 0.25 in a standard normal distribution table gives us an area of approximately 0.5987 to the left of 62. Multiplying this by the total population (0.5987 * 1500) gives approximately 898 values.

Question 17

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are greater than 54?

Accepted Answer

To find the number of values greater than 54, we first calculate the Z-score for 54 using the formula Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. For X = 54, μ = 60, and σ = 8, Z = (54 - 60) / 8 = -0.75. Looking up -0.75 on the Z-table or using a standard normal distribution calculator, we find that the proportion of values less than 54 is approximately 0.2266. Therefore, the proportion of values greater than 54 is 1 - 0.2266 = 0.7734. Multiplying this proportion by the total population of 1,500 gives 1,500 * 0.7734 ≈ 1,160.

Question 18

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are between 58 and 65?

Accepted Answer

The z-score for a value of 58 is (58 - 60) / 8 = -0.25. The z-score for a value of 65 is (65 - 60) / 8 = 0.625. The area under the normal curve between these two z-scores is 0.502, which corresponds to 502 values in the population.

Question 19

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. The 75th percentile most closely corresponds to which value of Z?

Accepted Answer

The 75th percentile in a normal distribution corresponds to a Z-score of approximately 0.674 (from Z-tables or standard normal distribution tables). To find the corresponding value in the population, use the formula: X = μ + Zσ, where μ is the mean, σ is the standard deviation, and Z is the Z-score. Substituting the given values: X = 60 + 0.674(8) = 60 + 5.392 = 65.392, which is closest to 65.4.

Question 20

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. What population value is larger than exactly 40% of the population?

Accepted Answer

The answer of Imagine a population consisting of 1,500 values...

Quiz 10: The Normal Distribution

The total area under the normal probability distribution is equal to which of the following?

For the equation for a normal distribution shown, what are the mean and variance? ? $f(x) =\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-16) ^{2}}{32}}$

For the equation for a normal distribution shown, what are the mean and standard deviation? ? $f(x) =\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-40) ^{2}}{32}}$

Which equation shown is correct for a normal distribution with a mean of 20 and a variance of 10?

Which equation shown is correct for a normal distribution with a median of 25 and a standard deviation of 5?

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

Which of the following is not a good rule of thumb for data values exhibiting a normal distribution?

For values from a data set, when we take the value, subtract the mean, and then divide by the standard deviation, we create new values called which of the following??

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are less than 93?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are greater than 84?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are between 87 and 99?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value of Z below most closely corresponds to the 65th percentile?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value below is closest to the value larger than exactly 55% of the population?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What proportion of values are larger than 78?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. Which value below is closest to the IQR?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are less than 62?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are greater than 54?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are between 58 and 65?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. The 75th percentile most closely corresponds to which value of Z?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. What population value is larger than exactly 40% of the population?

Quiz 10: The Normal Distribution

The total area under the normal probability distribution is equal to which of the following?

For the equation for a normal distribution shown, what are the mean and variance? ? f(x) =132πe−(x−16) 232f(x) =\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-16) ^{2}}{32}}f(x) =32π​1​e32−(x−16) 2​

For the equation for a normal distribution shown, what are the mean and standard deviation? ? f(x) =132πe−(x−40) 232f(x) =\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-40) ^{2}}{32}}f(x) =32π​1​e32−(x−40) 2​

Which equation shown is correct for a normal distribution with a mean of 20 and a variance of 10?

Which equation shown is correct for a normal distribution with a median of 25 and a standard deviation of 5?

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

Which of the following is not a good rule of thumb for data values exhibiting a normal distribution?

For values from a data set, when we take the value, subtract the mean, and then divide by the standard deviation, we create new values called which of the following??

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are less than 93?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are greater than 84?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are between 87 and 99?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value of Z below most closely corresponds to the 65th percentile?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value below is closest to the value larger than exactly 55% of the population?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What proportion of values are larger than 78?

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. Which value below is closest to the IQR?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are less than 62?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are greater than 54?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are between 58 and 65?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. The 75th percentile most closely corresponds to which value of Z?

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. What population value is larger than exactly 40% of the population?

For the equation for a normal distribution shown, what are the mean and variance? ? $f(x) =\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-16) ^{2}}{32}}$

For the equation for a normal distribution shown, what are the mean and standard deviation? ? $f(x) =\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-40) ^{2}}{32}}$