Chapter 6, The Normal Distribution Video Solutions, Introductory Statistics | Numerade (2024)

What is a density curve?

State the two basic properties of every density curve.

For a variable with a density curve, what is the relationship between the percentage of all possible observations of the variable that lie within any specified range and the corresponding area under its density curve?

Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct.
The percentage of all possible observations of the variable that lic between 7 and 12 cquals the area under its density curve between _____________ and ____________ expressed as a percentage.

Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct.
The percentage of all possible observations of the variable that lie to the right of 4 equals the area under its density curve to the right of _____________ expressed as a percentage.

Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct.
The area under the density curve that lies to the left of 10 is $0.654 .$ What percentage of all possible observations of the variable are
a. less than $10 ?$
b. at least $10 ?$

The area under the density curve that lies to the right of 15 is $0.324 .$ What percentage of all possible observations of the variable
a. exceed $15 ?$
b. are at most $15 ?$

Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct.
The area under the density curve that lies between 30 and 40 is $0.832 .$ What percentage of all possible observations of the variable are either less than 30 or greater than $40 ?$

Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct.
The area under the density curve that lics between 15 and 20 is $0.414 .$ What percentage of all possible observations of the variable are either less than 15 or greater than $20 ?$

Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct.
Given that $33.6 \%$ of all possible observations of the variable exceed $8,$ determine the area under the density curve that lies to the
a. right of $8 .$
b. left of 8

Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct.
Given that $28.4 \%$ of all possible observations of the variable are less than 11 , determine the area under the density curve that lies to the
a. left of $11 .$
b. right of 11

A curve has area 0.425 to the left of 4 and area 0.585 to the right of 4 . Could this curve be a density curve for some variable? Explain your answer.

A curve has area 0.613 to the left of 65 and area 0.287 to the right of $65 .$ Could this curve be a density curve for some variable? Explain your answer.

Explain in your own words why a density curve has the two properties listed in Key Fact 6.1 on page 254

A variable is approximately normally distributed. If you draw a histogram of the distribution of the variable, roughly what shape will it have?

Precisely what is meant by the statement that a population is normally distributed?Precisely what is meant by the statement that a population is normally distributed?

Two normally distributed variables have the same means and the same standard deviations. What can you say about their distributions? Explain your answer.

Which normal distribution has a wider spread: the one with mean 1 and standard deviation 2 or the one with mean 2 and standard deviation 1? Explain your answer.

Consider two normal distributions, one with mean -4 and standard deviation $3,$ and the other with mean 6 and standard deviation $3 .$ Answer true or false to each statement and explain your answers.
a. The two normal distributions have the same shape.
b. The two normal distributions are centered at the same place.

Consider two normal distributions, one with mean -4 and standard deviation $3,$ and the other with mean -4 and standard deviation $6 .$ Answer true or false to each statement and explain
your answers.
a. The two normal distributions have the same shape.
b. The two normal distributions are centered at the same place.

True or false: The mean of a normal distribution has no effect on its shape. Explain your answer.

What are the parameters for a normal curve?

Sketch the normal distribution with
a. $\mu=3$ and $\sigma=3$
b. $\mu=1$ and $\sigma=3$
c. $\mu=3$ and $\sigma=1$

Sketch the normal distribution with
a. $\mu=-2$ and $\sigma=2$
b. $\mu=-2$ and $\sigma=1 / 2$
c. $\mu=0$ and $\sigma=2$

For a normally distributed variable, what is the relationship between the percentage of all possible observations that lie between 2 and 3 and the area under the associated normal curve between 2 and $3 ?$ What if the variable is only approximately normally distributed?

For a normally distributed variable, what is the relationship between the percentage of all possible observations that lie to the right of 7 and the area under the associated normal curve to the right of $7 ?$ What if the variable is only approximately normally distributed?

The area under a particular normal curve to the left of 105 is $0.6227 .$ A normally distributed variable has the same mean and standard deviation as the parameters for this normal curve. What percentage of all possible observations of the variable lie to the Ieft of $105 ?$ Explain your answer.

The area under a particular normal curve between 10 and 15 is $0.6874 .$ A normally distributed variable has the same mean and standard deviation as the parameters for this normal curve. What percentage of all possible observations of the variable lie between 10 and $15 ?$ Explain your answer.

Female College Students. Refer to Example 6.1 on page 256
a. Use the relative-frequency distribution in Table 6.1 to obtain the percentage of female students who are between 60 and 65 inches tall.
b. Use your answer from part (a) to estimate the area under the normal curve having parameters $\mu=64.4$ and $\sigma=2.4$ that lies between 60 and $65 .$ Why do you get only an estimate of the true area?

Female College Students. Refer to Example 6.1 on page 256
a. The area under the standard normal curve with parameters $\mu=64.4$ and $\sigma=2.4$ that lies to the left of 61 is $0.0783 .$ Use this information to estimate the percentage of female students who are shorter than 61 inches.
b. Use the relative-frequency distribution in Table 6.1 to obtain the exact percentage of female students who are shorter than
61 inches.
c. Compare your answers from parts (a) and (b).

Giant Tarantulas. One of the larger species of tarantulas is the Grammostola mollicoma, whose common name is the Brazilian giant tawny red. A tarantula has two body parts. The anterior part of the body is covered above by a shell, or carapace. From a recent article by F. costa and F. Perez-Miles titled "Reproductive Biology of Uruguayan Theraphosids" (The Journal of Arachnology, Vol. 30. No. 3, pp. 571-587), we find that the carapace length of the adult male $G$. mollicoma is normally distributed with a mean of $18.14 \mathrm{mm}$ and a standard deviation of $1.76 \mathrm{mm}$. Let $x$ denote carapace length for the adult male G. mollicoma.
a. Sketch the distribution of the variable $x .$
b. Obtain the standardized version, $z,$ of $x$
c. Identify and sketch the distribution of $z$
d. The percentage of adult male $G .$ mollicoma that have carapace length between $16 \mathrm{mm}$ and $17 \mathrm{mm}$ is equal to the area under the standard normal curve between _______ and _______.
e. The percentage of adult male $G$. mollicoma that have carapace length excecding $19 \mathrm{mm}$ is equal to the area under the standard normal curve that lies to the ___________ of ____________.

Serum Cholesterol Levels. According to the National Health and Nutrition Examination Survey, published by the National Center for Health Statistics, the serum (noncellular portion of blood) total cholesterol level of U.S. females 20 years old or older is normally distributed with a mean of 206 mg/dL (milligrams per deciliter) and a standard deviation of $44.7 \mathrm{mg} / \mathrm{dL}$ Let $x$ denote scrum total cholesterol level for U.S. females
20 years old or older.
a. Sketch the distribution of the variable $x$.
b. Obtain the standardized version, $z$. of $x$.
c. Identify and skeich the distribution of $z$
d. The percentage of U.S. females 20 years old or older who have a serum total cholesterol level between 150 mg/dL. and $250 \mathrm{mg} / \mathrm{dL}$ is equal to the area under the standard normal curve between ________ and ________.
e. The percentage of U.S. females 20 years old or older who have a serum total cholesterol level below $220 \mathrm{mg} / \mathrm{dL}$, is equal to the area under the standard normal curve that lics to the __________ of _________

New York City 10-km Run. As reported in Runner's World magazine, the times of the finishers in the New York City $10-\mathrm{km}$ run are normally distributed with mean 61 minutes and standard deviation 9 minutes. Let $x$ denote finishing time for finishers in this race.
a. Sketch the distribution of the variable $x$.
b. Obtain the standardized version, $z,$ of $x$
c. Identify and sketch the distribution of $z$
d. The percentage of finishers with times between 50 and 70 minutes is equal to the area under the standard normal curve between _________ and ___________.

Green Sea Urchins. From the paper "Effects of Chronic Nitrate Exposure on Gonad Growth in Green Sea Urchin Strongylocentrotus droebachiensis" (Aquaculture, Vol. $242,$ No. $1-4$ pp. $357-363$ ) by S. Siikavuopio et al., we found that weights of adult green sea urchins are normally distributed with mean $52.0 \mathrm{g}$ and standard deviation $17.2 \mathrm{g}$. Let $x$ denote weight of adult green sea urchins.
a. Sketch the distribution of the variable $x$.
b. Obtain the standardized version, $z,$ of $x$
c. Identify and sketch the distribution of $z$
d. The percentage of adult green sea urchins with weights between $50 \mathrm{g}$ and $60 \mathrm{g}$ is equal to the area under the standard normal curve between _________ and _________.
e. The percentage of adult green sea urchins with weights above $40 \mathrm{g}$ is equal to the area under the standard normal curve that lies to the ________ of __________.

Ages of Mothers. From the document National Vital Statistics Reports, a publication of the National Center for Health Statistics, we obtained the following frequency distribution for the ages of women who became mothers during one year.
$$\begin{array}{c|c}
\hline \text { Age }(\mathrm{yr}) & \text { Frequency } \\
\hline 10-\text { under } 15 & 7,315 \\
15 \text { -under } 20 & 425,493 \\
20 \text { -under } 25 & 1,022,106 \\
25 \text { -under } 30 & 1,060,391 \\
30 \text { -under } 35 & 951,219 \\
35 \text { -under } 40 & 453,927 \\
40 \text { -under } 45 & 95,788 \\
45 \text { -under } 50 & 54,872 \\
\hline
\end{array}$$
a. Obtain a relative-frequency histogram of these age data.
b. Based on your histogram, do you think that the ages of women who became mothers that year are approximately normally distributed? Explain your answer.

Birth Rates. The National Center for Health Statistics publishes information about birth rates (per 1000 population) in the document National Vital Statistics Report. The following table provides a frequency distribution for birth rates during one year for the 50 states and the District of Columbia.
$$\begin{array}{|c|c||c|c|}
\hline \text { Rate } & \text { Frequency } & \text { Rate } & \text { Frequency } \\
\hline 10 \text { -under } 11 & 2 & 16 \text { -under } 17 & 1 \\
11 \text { -under } 12 & 3 & 17 \text { -under } 18 & 1 \\
12 \text { -under } 13 & 10 & 18 \text { -under } 19 & 0 \\
13 \text { -under } 14 & 17 & 19 \text { -under } 20 & 0 \\
14 \text { -under } 15 & 9 & 20-\text { under } 21 & 0 \\
15 \text { -under } 16 & 7 & 21 \text { -under } 22 & 1 \\
\hline
\end{array}$$
a. Obtain a frequency histogram of these birth-rate data.
b. Based on your histogram, do you think that birth rates for the 50 states and the District of Columbia are approximately normally distributed? Explain your answer.

Cloudiness in Bresiau. In the paper "Cloudiness: Note on a Novel Case of Frequency" (Proceedings of the Royal Society of London, VoL. $62,$ pp. $287-290$ ), K. Pearson examined data on daily degree of cloudiness, on a scale of 0 to $10,$ at Breslau (Wroclaw), Poland, during the decade $1876-1885 .$ A frequency distribution of the data is presented in the following table.
$$\begin{array}{|c|c||c|c|}
\hline \text { Degree } & \text { Frequency } & \text { Degree } & \text { Frequency } \\
\hline 0 & 751 & 6 & 21 \\
1 & 179 & 7 & 71 \\
2 & 107 & 8 & 194 \\
3 & 69 & 9 & 117 \\
4 & 46 & 10 & 2089 \\
5 & 9 & & \\
\hline
\end{array}$$
a. Draw a frequency histogram of these degree-of-cloudiness data.
b. Based on your histogram, do you think that degree of cloudiness in Breslau during the decade in question is approximately normally distributed? Explain your answer.

Wrong Number. A classic study by F. Thorndike on the number of calls to a wrong number appeared in the paper "Applications of Poisson's Probability Summation" (Bell Systems Technical Joumal, Vol. $5,$ pp. $604-624$ ). The study examined the number of calls to a wrong number from coin-box telephones in a large transportation terminal. Based on the results of that paper, we obtained the following percent distribution for the number of wrong numbers during a 1 -minute period.
$$\begin{array}{|l|ccccccccc}
\hline \text { Wrong } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline \text { Percent } & 17.2 & 30.5 & 26.6 & 15.1 & 7.3 & 2.4 & 0.7 & 0.1 & 0.1 \\
\hline
\end{array}$$
a. Construct a relative-frequency histogram of these wrongnumber data.
b. Based on your histogram, do you think that the number of wrong numbers from these coin-box telephones is approximately normally distributed? Explain your answer.

SAT Scores. Each year, thousands of high school students bound for college take the Scholastic Assessment Test (SAT). This test measures the verbal and mathematical abilities of prospective college students. Student scores are reported on a scale that ranges from a low of 200 to a high of $800 .$ Summary results for the scores are published by the College Entrance Examination Board in College Bound Seniors. In one high school graduating class, the SAT scores are as provided on the WeissStats CD. Use the technology of your choice to answer the following questions.
a. Do the SAT verbal scores for this class appear to be approximately normally distributed? Explain your answer.
b. Do the SAT math scores for this class appear to be approximately normally distributed? Explain your answer.

Fertility Rates. From the U.S. Census Bureau, in the document International Data Base, we obtained data on the total fertility rates for women in various countries. Those data are presented on the WeissStats CD. The total fertility rate gives the average number of children that would be born if all women in a given country lived to the end of their childbearing years and, at each year of age, they experienced the birth rates occurring in the specified year. Use the technology of your choice to decide whether total fertility rates for countries appear to be approximately normally distributed. Explain your answer.

"Chips Ahoy! 1,000 Chips Challenge"" Students in an introductory statistics course at the U.S. Air Force Academy participated in Nabisco's "Chips Ahoy! 1,000 Chips Challenge" by confirming that there were at least 1000 chips in every 18 -ounce bag of cookies that they examined. As part of their assignment, they concluded that the number of chips per bag is approximately normally distributed. Could the number of chips per bag be exactly normally distributed? Explain your answer. [SOURCE:
B. Warner and J. Rutledge, "Checking the Chips Ahoy! Guarantee," Chance, Vol. $12(1), \text { pp. } 10-14]$

Consider a normal distribution with mean 5 and standard deviation 2
a. Sketch the associated normal curve.
b. Use the footnote on page 255 to write the equation of the associated normal curve.
c. Use the technology of your choice to graph the equation obtained in part (b).
d. Compare the curves that you obtained in parts (a) and (c).

Gestation Periods of Humans. Refer to the simulation of human gestation periods discussed in Example 6.2 on page 259
a. Sketch the normal curve for human gestation periods.
b. Simulate 1000 human gestation periods. (Note: Users of the TI-83/84 Plus should simulate 500 human gestation periods.)
c. Approximately what values would you expect for the sample mean and sample standard deviation of the 1000 observations? Explain your answers.
d. Obtain the sample mean and sample standard deviation of the 1000 observations, and compare your answers to your estimates in part (c).
e. Roughly what would you expect a histogram of the 1000 observations to look like? Explain your answer.
f. Obtain a histogram of the 1000 observations, and compare your result to your expectation in part (e).

Delaying Adulthood. In the paper, "Delayed Metamorphosis of a Tropical Reef Fish (Acanthurus triostegus): A Field Experiment" (Marine Ecology Progress Series. Vol. 176, pp. $25-38$ ), M. McCormick studied larval duration of the convict surgeonfish, a common tropical reef fish. This fish has been found to delay metamorphosis into adulthood by extending its larval phase, a delay that often leads to enhanced survivorship in the species by increasing the chances of finding suitable habitat. Duration of the larval phase for convict surgeonfish is normally distributed with mean 53 days and standard deviation 3.4 days. Let $x$ denote larval-phase duration for convict surgeonfish.
a. Sketch the normal curve for the variable $x$.
b. Simulate 1500 observations of $x$. (Note: Users of the TI-83/84 Plus should simulate 750 observations.)
c. Approximately what values would you expect for the sample mean and sample standard deviation of the 1500 observations? Explain your answers.
d. Obtain the sample mean and sample standard deviation of the 1500 observations, and compare your answers to your estimates in part (c).
e. Roughly what would you expect a histogram of the 1500 observations to look like? Fxplain your answer.
f. Obtain a histogram of the 1500 observations, and compare your result to your expectation in part (e).