Chapter 6, Normal Probability Distributions Video Solutions, Essentials of Statistics | Numerade (2024)

What's wrong with the following statement? "Because the digits 0,1 , $2, \ldots, 9$ are the normal results from lottery drawings, such randomly selected numbers have a normal distribution."

A normal distribution is informally described as a probability distribution that is bell-shaped when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.

Identify the two requirements necessary for a normal distribution to be a standard normal distribution.

What does the notation $z_{\alpha}$ indicate?

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Greater than $3.00$ minutes

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Less than $4.00$ minutes

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Between 2 minutes and 3 minutes

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Between $2.5$ minutes and $4.5$ minutes

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=0.44
$$

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=-1.04
$$

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=-0.84 \quad z=1.28
$$

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=-1.07 \quad z=0.67
$$

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.8907
$$

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.3050
$$

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.9265
$$

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.2061
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than }-1.23
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than }-1.96
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 1.28
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 2.56
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than } 0.25
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than } 0.18
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than }-2.00
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than }-3.05
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between } 2.00 \text { and } 3.00
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between } 1.50 \text { and } 2.50
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between and }-2.55 \text { and }-2.00
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-2.75 \text { and }-0.75
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-2.00 \text { and } 2.00
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-3.00 \text { and } 3.00
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-1.00 \text { and } 5.00
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-4.27 \text { and } 2.34
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 4.55
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than }-3.75
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than } 0
$$

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 0 \text { . }
$$

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
Find $P_{99}$, the 99 th percentile. This is the bone density score separating the bottom $99 \%$ from the top $1 \%$.

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
Find $P_{10}$, the 10 th percentile. This is the bone density score separating the bottom $10 \%$ from the top $90 \%$.

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
If bone density scores in the bottom $2 \%$ and the top $2 \%$ are used as cutoff points for levels that are too low or too high, find the two readings that are cutoff values.

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
Find the bone density scores that can be used as cutoff values separating the lowest $3 \%$ and highest $3 \%$.

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.10}
$$

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.02}
$$

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.04}
$$

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.15}
$$

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About __________$\%$ of the area is between $z=-1$ and $z=1$ (or within 1 standard deviation
of the mean).

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About ________ $\%$ of the area is between $z=-2$ and $z=2$ (or within 2 standard deviations of the mean).

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About _______ $\%$ of the area is between $z=-3$ and $z=3$ (or within 3 standard deviations of the mean).

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About $\quad \%$ of the area is between $z=-3.5$ and $z=3.5$ (or within $3.5$ standard deviations of the mean).

For bone density scores that are normally distributed with a mean of 0 and a standard deviation of 1 , find the percentage of scores that are
a. significantly high (or at least 2 standard deviations above the mean).
b. significantly low (or at least 2 standard deviations below the mean).
c. not significant (or less than 2 standard deviations away from the mean).

In a continuous uniform distribution,
$$
\mu=\frac{\text { minimum }+\text { maximum }}{2} \text { and } \sigma=\frac{\text { range }}{\sqrt{12}}
$$
a. Find the mean and standard deviation for the distribution of the waiting times represented in Figure $6-2$, which accompanies Exercises $5-8$.
b. For a continuous_uniform distribution with $\mu=0$ and $\sigma=1$, the minimum is $-\sqrt{3}$ and the maximum is $\sqrt{3}$. For this continuous uniform distribution, find the probability of randomly selecting a value between $-1$ and 1 , and compare it to the value that would be obtained by incorrectly treating the distribution as a standard normal distribution. Does the distribution affect the results very much?