Question 1

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve to the right of $z = - 1.25$.

Accepted Answer

The area to the right of $z = -1.25$ on the standard normal curve is equivalent to 1 minus the area to the left of $z = -1.25$. The area to the left of $z = -1.25$ is approximately 0.1056 (from standard normal distribution tables or a calculator). Therefore, the area to the right is $1 - 0.1056 = 0.8944$.

Question 2

Explain the Role of Area in the Normal Density Function
-The area under the normal curve drawn with regard to the population parameters is the same as the proportion of the population that has these characteristics.

Accepted Answer

The area under the normal curve represents the probability of a random variable falling within a particular range. Therefore, when the curve is drawn with respect to population parameters, the area under the curve between any two points corresponds to the proportion of the population that falls within that range, reflecting those characteristics.

Question 3

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve between $z = - 1.5$ and $z = 2.5$.

Accepted Answer

The area under the standard normal curve between $z = -1.5$ and $z = 2.5$ can be found by looking up the values in a standard normal distribution table or using a calculator with statistical functions. The area to the left of $z = 2.5$ is approximately 0.9938, and the area to the left of $z = -1.5$ is approximately 0.0668. Subtracting these values gives us the area between these two z-scores: $0.9938 - 0.0668 = 0.9270$.

Question 4

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve to the left of $\mathrm { z } = 1.5$.

Accepted Answer

The area under the standard normal curve to the left of $z = 1.5$ is found using standard normal distribution tables or a calculator designed for statistical functions, which typically gives a value of approximately $0.9332$. This represents the proportion of data under the curve to the left of the specified z-score in a standard normal distribution.

Question 5

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is
as given.
-$$X = 72 ; \sigma = 3$$

Accepted Answer

The z-score is calculated using the formula $$z = \frac{X - \mu}{\sigma}$$, where $X$ is the value from the dataset, $\mu$ is the mean, and $\sigma$ is the standard deviation. Plugging in the given values, we get $$z = \frac{72 - 60}{3} = \frac{12}{3} = 4$$.

Question 6

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve between $\mathrm { z } = - 1.25$ and $\mathrm { z } = 1.25$.

Accepted Answer

The area under the standard normal curve between $z = -1.25$ and $z = 1.25$ represents the probability that a standard normal variable falls within this range. This can be found using standard normal distribution tables or a calculator designed for statistical functions. The correct answer is approximately 0.7888, which means about 78.88% of the distribution falls within this range.

Question 7

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is
as given.
-Suppose a random variable $x$ is normally distributed with mean, $\mu$, and standard deviation, $\sigma$. Then the random variable $z = \frac { \bar { x } - \mu } { \sigma }$ has the graph of the

Accepted Answer

The formula given converts any normally distributed variable into a standard normal distribution, which has a mean of 0 and a standard deviation of 1.

Question 8

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve between $\mathrm { z } = 1.5$ and $\mathrm { z } = 2.5$.

Accepted Answer

The area under the standard normal curve between $z = 1.5$ and $z = 2.5$ can be found by looking up the values in a standard normal distribution table or using a calculator designed for statistical analysis. The area to the left of $z = 1.5$ is approximately $0.9332$, and the area to the left of $z = 2.5$ is approximately $0.9938$. Subtracting these values gives the area between $z = 1.5$ and $z = 2.5$, which is $0.9938 - 0.9332 = 0.0606$.

Question 9

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is
as given.
-$$X = 60 ; \sigma = 7$$

Accepted Answer

The z-score is calculated using the formula $z = \frac{X - \mu}{\sigma}$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Substituting $X = 60$, $\mu = 60$, and $\sigma = 7$, we get $z = \frac{60 - 60}{7} = 0$.

Question 10

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is
as given.
-$$X = 50 ; \sigma = 10$$

Accepted Answer

The z-score is calculated using the formula $z = \frac{X - \mu}{\sigma}$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Substituting the given values, $z = \frac{50 - 60}{10} = -1$.

Question 11

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve between $\mathrm { z } = 1$ and $\mathrm { z } = 2$.

Accepted Answer

The area under the standard normal curve between $z = 1$ and $z = 2$ is found by subtracting the area from $z = 0$ to $z = 1$ from the area from $z = 0$ to $z = 2$. Using standard normal distribution tables or a calculator, the area from $z = 0$ to $z = 1$ is approximately $0.3413$, and the area from $z = 0$ to $z = 2$ is approximately $0.4772$. Subtracting these gives $0.4772 - 0.3413 = 0.1359$.

Question 12

Explain the Role of Area in the Normal Density Function
-The proportion of the population that has certain characteristics is the same as the probability that a randomly selected individual of the population has these same characteristics.

Accepted Answer

In the context of the normal density function, the area under the curve corresponding to a specific range of values represents the proportion of the population that falls within that range. This proportion is equivalent to the probability that a randomly selected individual from the population will have characteristics (e.g., measurements, scores) that fall within the specified range.

Question 13

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is
as given.
-$$X = 68 ; \sigma = 8$$

Accepted Answer

The z-score is calculated using the formula $z = \frac{X - \mu}{\sigma}$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Substituting the given values, $z = \frac{68 - 60}{8} = 1$.

Question 14

Explain the Role of Area in the Normal Density Function
-The area under the normal curve drawn with regard to the population parameters is the same as the probability that a randomly selected individual of a population has these characteristics.

Accepted Answer

The area under the normal curve represents the probability of a randomly selected individual from the population falling within a specific range of values. This is a fundamental concept in statistics, where the total area under the curve equals 1, or 100% probability.

Question 15

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve to the right of $\mathrm { z } = 1$.

Accepted Answer

The area to the right of $z = 1$ on the standard normal curve is the complement of the area to the left of $z = 1$, which is typically found in a standard normal distribution table or calculated using statistical software. The area to the left of $z = 1$ is approximately $0.8413$, so the area to the right is $1 - 0.8413 = 0.1587$.

Question 16

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve between $\mathrm { z } = 0$ and $\mathrm { z } = 3$.

Accepted Answer

The area under the standard normal curve between $ z = 0 $ and $ z = 3 $ is approximately 0.4987, as found using standard normal distribution tables or a calculator.

Question 17

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is
as given.
-$$X = 45 ; \sigma = 4$$

Accepted Answer

The z-score is calculated using the formula $ z = \frac{X - \mu}{\sigma} $. Here, $ X = 45 $, $ \mu = 60 $, and $ \sigma = 4 $. So, $ z = \frac{45 - 60}{4} = -3.75 $.

Question 18

Find the Area under the Standard Normal Curve
-Find the sum of the areas under the standard normal curve to the left of $\mathrm { z } = - 1.25$ and to the right of $\mathrm { z } = 1.25$.

Accepted Answer

The answer of Find the Area under the Standard Normal...

Question 19

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is
as given.
-$$X = 66 ; \sigma = 4$$

Accepted Answer

The z-score is calculated using the formula $z = \frac{X - \mu}{\sigma}$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Substituting the given values, $z = \frac{66 - 60}{4} = 1.5$.

Question 20

Find the Area under the Standard Normal Curve
-Find the area under the standard normal curve to the left of $\mathrm { z } = 1.25$.

Accepted Answer

The area under the standard normal curve to the left of $z = 1.25$ is found using the standard normal distribution table or a calculator designed for statistical functions. This area represents the probability that a standard normal random variable takes on a value less than $1.25$, which is $0.8944$.

Quiz 7: The Normal Probability Distribution

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the right of $z = - 1.25$ .

Explain the Role of Area in the Normal Density Function -The area under the normal curve drawn with regard to the population parameters is the same as the proportion of the population that has these characteristics.

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $z = - 1.5$ and $z = 2.5$ .

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the left of $\mathrm { z } = 1.5$ .

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - $X = 72 ; \sigma = 3$

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $\mathrm { z } = - 1.25$ and $\mathrm { z } = 1.25$ .

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $\mathrm { z } = 1.5$ and $\mathrm { z } = 2.5$ .

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - $X = 60 ; \sigma = 7$

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - $X = 50 ; \sigma = 10$

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $\mathrm { z } = 1$ and $\mathrm { z } = 2$ .

Explain the Role of Area in the Normal Density Function -The proportion of the population that has certain characteristics is the same as the probability that a randomly selected individual of the population has these same characteristics.

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - $X = 68 ; \sigma = 8$

Explain the Role of Area in the Normal Density Function -The area under the normal curve drawn with regard to the population parameters is the same as the probability that a randomly selected individual of a population has these characteristics.

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the right of $\mathrm { z } = 1$ .

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $\mathrm { z } = 0$ and $\mathrm { z } = 3$ .

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - $X = 45 ; \sigma = 4$

Find the Area under the Standard Normal Curve -Find the sum of the areas under the standard normal curve to the left of $\mathrm { z } = - 1.25$ and to the right of $\mathrm { z } = 1.25$ .

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - $X = 66 ; \sigma = 4$

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the left of $\mathrm { z } = 1.25$ .

Quiz 7: The Normal Probability Distribution

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the right of z=−1.25z = - 1.25z=−1.25 .

Explain the Role of Area in the Normal Density Function -The area under the normal curve drawn with regard to the population parameters is the same as the proportion of the population that has these characteristics.

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between z=−1.5z = - 1.5z=−1.5 and z=2.5z = 2.5z=2.5 .

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the left of z=1.5\mathrm { z } = 1.5z=1.5 .

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - X=72;σ=3X = 72 ; \sigma = 3X=72;σ=3

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between z=−1.25\mathrm { z } = - 1.25z=−1.25 and z=1.25\mathrm { z } = 1.25z=1.25 .

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between z=1.5\mathrm { z } = 1.5z=1.5 and z=2.5\mathrm { z } = 2.5z=2.5 .

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - X=60;σ=7X = 60 ; \sigma = 7X=60;σ=7

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - X=50;σ=10X = 50 ; \sigma = 10X=50;σ=10

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between z=1\mathrm { z } = 1z=1 and z=2\mathrm { z } = 2z=2 .