Question 1

A chemical manufacturer sells its product in steel drums. The net weight of the chemical in the drums has a normal distribution with mean 310 pounds and standard deviation 4 pounds. The manufacturer guarantees its customers each drum will contain at least 300 pounds of the chemical.
What percentage of the drums will satisfy this guarantee?
______________

Accepted Answer

The answer of A chemical manufacturer sells its product in...

Question 2

EPA studies of fuel consumption indicate compact car mileage to be normally distributed with a mean of 25.5 miles per gallon and a standard deviation of 5.0 mpg.
What percentage of compact cars gets between 20 and 40 mpg?
______________

Accepted Answer

The answer of EPA studies of fuel consumption indicate compact...

Question 3

If the z-value for a given value x of the random variable X is z = 1.96, and the distribution of X is normally distributed with a mean of 60 and a standard deviation of 6, to what x-value does this z-value correspond?

Accepted Answer

The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Plugging in the given values: $1.96 = \frac{x - 60}{6}$. Solving for $x$ gives $x = 1.96 \times 6 + 60 = 11.76 + 60 = 71.76$.

Question 4

For some value of z, the probability that a standard normal variable is below z is 0.2090. The value of z is:

Accepted Answer

The value of z that corresponds to a probability of 0.2090 in a standard normal distribution is approximately -0.81. This is found by looking up the probability in a standard normal (Z) table or using a statistical software to find the Z-score that corresponds to the lower tail probability of 0.2090.

Question 5

If Z is a standard normal random variable, the area between z = 0.0 and z = 1.50 compared to the area between z = 1.5 and z = 3.0 will be:

Accepted Answer

The area between z = 0.0 and z = 1.50 is larger because the standard normal distribution is more dense closer to the mean (z = 0). As you move further away from the mean, the density decreases, making the area between z = 1.5 and z = 3.0 smaller.

Question 6

For some positive value of x, the probability that a standard normal variable is between 0 and + 2x is 0.1255. The value of x is:

Accepted Answer

The answer of For some positive value of x, the...

Question 7

The distribution of IQ scores for high school graduates is normally distributed with  = 104 and  = 16.
Find the probability a person chosen at random from this group has an IQ score above 146.08.
______________
What fraction of the IQ scores would be between 97 and 126?
______________
What is the 95th percentile of this normal distribution?
______________

Accepted Answer

The answer of The distribution of IQ scores for high...

Question 8

For some positive value of x, the probability that a standard normal variable is between 0 and + 1.5x is 0.4332. The value of x is:

Accepted Answer

The probability of a standard normal variable being between 0 and +1.5x is given as 0.4332. In a standard normal distribution, the probability of being between 0 and +1 is approximately 0.3413. Given that 1.5x results in a probability of 0.4332, it suggests that x must be 1.00 to align with the standard normal distribution properties, where an increase in the standard deviation (x in this case) directly affects the probability range.

Question 9

The probability density function f(x) of a random variable X that is normally distributed is completely determined once the:

Accepted Answer

The probability density function of a normally distributed random variable is determined by its mean and standard deviation. The mean specifies the center of the distribution, and the standard deviation influences the spread of the distribution. Therefore, the correct answer is D, where both the mean and standard deviation are specified. The other choices are incorrect because they do not fully determine the probability density function of a normally distributed random variable.

Question 10

Let z denote a standard normal random variable. Find P(z > 1.48). ______________ Find P(-0.44 < z < 2.68). ______________ Determine the value of z₀ which satisfies P(z z₀) = 0.7995. ______________ Find P(z < -0.87). ______________ Find P(-1.66 < z -0.48). ______________ Find z0 such that P(-z₀ < z < z₀) = 0.901. ______________ Find z0 such that P(z < z₀) = 0.0375. ______________ Find z0 such that P(-z₀ < z < z₀) = 0.7698. ______________

Accepted Answer

The answer of Let z denote a standard normal random...

Question 11

If Z is a standard normal random variable, the area between z = 0.0 and z = 1.30 is 0.4032, while the area between z = 0.0 and z = 1.50 is 0.4332. What is the area between z = -1.30 and z = 1.50?

Accepted Answer

The area between z = -1.30 and z = 1.50 is the sum of the area from z = 0 to z = 1.30 and the area from z = 0 to z = 1.50, because the standard normal distribution is symmetric about z = 0. Therefore, the total area is 0.4032 (from 0 to 1.30) + 0.4332 (from 0 to 1.50) = 0.8364.

Question 12

Which of the following is not true for a normal distribution?

Accepted Answer

A normal distribution is continuous, not discrete.

Question 13

Which of the following distributions are always symmetrical?

Accepted Answer

The normal distribution is always symmetrical. The exponential distribution is always skewed to the right. The binomial distribution is usually skewed, except when the number of trials is very large and/or the probability of success is close to 0.5. The statement "All continuous distributions are symmetrical" is false.

Question 14

What proportion of the data from a normal distribution is within two standard deviations from the mean?

Accepted Answer

Approximately 95.44% of the data from a normal distribution is within two standard deviations from the mean, which corresponds to choice D (0.9544).

Question 15

Which of the following distributions is considered the cornerstone distribution of statistical inference?

Accepted Answer

The normal distribution is considered the cornerstone distribution of statistical inference because it has several properties that make it convenient for making statistical inferences. It is symmetric, unimodal, and bell-shaped, and it has well-defined parameters that can be used for hypothesis testing and estimation. Additionally, many statistical methods and tests, such as the t-test and ANOVA, assume that data are normally distributed.

Question 16

Given that Z is a standard normal random variable, the area to the left of a value z is expressed as:

Accepted Answer

The area to the left of a value z corresponds to the cumulative probability up to z, which is expressed as P(Z ≤ z) for a standard normal distribution.

Question 17

For some positive value of z, the probability that a standard normal variable is between 0 and z is 0.3770. The value of z is:

Accepted Answer

Using a standard normal distribution table, we can find the z-score that corresponds to a probability of 0.3770. This z-score is approximately 1.16. So the answer to this question is choice C).

Question 18

Two students are enrolled in an introductory statistics class at the university. The first student is in a morning section and the second student is in an afternoon section. Assume test scores are normally distributed. In the morning section, the mean was 64 and the standard deviation was 8. In the afternoon section, the class mean was 60 with a standard deviation of 7.5. If the student in the morning section takes a midterm exam and earns a score of 76, while the student in the afternoon section takes a midterm exam and earns a score of 72, which student performed better compared to the rest of the students in his or her respective class?
______________
Explain.
________________________________________________________

Accepted Answer

The answer of Two students are enrolled in an introductory...

Question 19

A random variable x is normally distributed with m = 100 and  = 20.
What is the median of this distribution?
______________
Find  .
______________
Find  .
______________

Accepted Answer

The answer of A random variable x is normally distributed...

Question 20

The mean SAT verbal score of next year's entering freshmen at the local college is 600. The college also knows 69.5% of the students scored less than 625. If the scores are normally distributed, what is the standard deviation of the SAT scores?
______________

Accepted Answer

The answer of The mean SAT verbal score of next...

Quiz 6: The Normal Probability Distribution

EPA studies of fuel consumption indicate compact car mileage to be normally distributed with a mean of 25.5 miles per gallon and a standard deviation of 5.0 mpg. What percentage of compact cars gets between 20 and 40 mpg? ______________

If the z-value for a given value x of the random variable X is z = 1.96, and the distribution of X is normally distributed with a mean of 60 and a standard deviation of 6, to what x-value does this z-value correspond?

For some value of z, the probability that a standard normal variable is below z is 0.2090. The value of z is:

If Z is a standard normal random variable, the area between z = 0.0 and z = 1.50 compared to the area between z = 1.5 and z = 3.0 will be:

For some positive value of x, the probability that a standard normal variable is between 0 and + 2x is 0.1255. The value of x is:

For some positive value of x, the probability that a standard normal variable is between 0 and + 1.5x is 0.4332. The value of x is:

The probability density function f(x) of a random variable X that is normally distributed is completely determined once the:

If Z is a standard normal random variable, the area between z = 0.0 and z = 1.30 is 0.4032, while the area between z = 0.0 and z = 1.50 is 0.4332. What is the area between z = -1.30 and z = 1.50?

Which of the following is not true for a normal distribution?

Which of the following distributions are always symmetrical?

What proportion of the data from a normal distribution is within two standard deviations from the mean?

Which of the following distributions is considered the cornerstone distribution of statistical inference?

Given that Z is a standard normal random variable, the area to the left of a value z is expressed as:

For some positive value of z, the probability that a standard normal variable is between 0 and z is 0.3770. The value of z is:

A random variable x is normally distributed with m = 100 and = 20. What is the median of this distribution? Find . Find . __

The mean SAT verbal score of next year's entering freshmen at the local college is 600. The college also knows 69.5% of the students scored less than 625. If the scores are normally distributed, what is the standard deviation of the SAT scores? ______________

Quiz 6: The Normal Probability Distribution

EPA studies of fuel consumption indicate compact car mileage to be normally distributed with a mean of 25.5 miles per gallon and a standard deviation of 5.0 mpg. What percentage of compact cars gets between 20 and 40 mpg? ______________

If the z-value for a given value x of the random variable X is z = 1.96, and the distribution of X is normally distributed with a mean of 60 and a standard deviation of 6, to what x-value does this z-value correspond?

For some value of z, the probability that a standard normal variable is below z is 0.2090. The value of z is:

If Z is a standard normal random variable, the area between z = 0.0 and z = 1.50 compared to the area between z = 1.5 and z = 3.0 will be:

For some positive value of x, the probability that a standard normal variable is between 0 and + 2x is 0.1255. The value of x is:

For some positive value of x, the probability that a standard normal variable is between 0 and + 1.5x is 0.4332. The value of x is:

The probability density function f(x) of a random variable X that is normally distributed is completely determined once the:

If Z is a standard normal random variable, the area between z = 0.0 and z = 1.30 is 0.4032, while the area between z = 0.0 and z = 1.50 is 0.4332. What is the area between z = -1.30 and z = 1.50?

Which of the following is not true for a normal distribution?

Which of the following distributions are always symmetrical?

What proportion of the data from a normal distribution is within two standard deviations from the mean?

Which of the following distributions is considered the cornerstone distribution of statistical inference?

Given that Z is a standard normal random variable, the area to the left of a value z is expressed as:

For some positive value of z, the probability that a standard normal variable is between 0 and z is 0.3770. The value of z is:

A random variable x is normally distributed with m = 100 and = 20. What is the median of this distribution? ______________ Find . ______________ Find . ______________

The mean SAT verbal score of next year's entering freshmen at the local college is 600. The college also knows 69.5% of the students scored less than 625. If the scores are normally distributed, what is the standard deviation of the SAT scores? ______________

A random variable x is normally distributed with m = 100 and = 20. What is the median of this distribution? Find . Find . __