The Normal Distribution

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

True or False:

The mean of a normal distribution has no effect on its spread. Explain your answer.

What are the parameters for a normal curve?

Sketch the normal distribution with 

$$\begin{gathered} \left(a\right) \mu =3; \sigma =3 \\ \left(b\right) \mu =1; \sigma =3 \\ \left(c\right) \mu =3; \sigma =1 \end{gathered}$$

Sketch the normal distribution with 

$$\begin{gathered} \left(a\right) \mu =-2; \sigma =2 \\ \left(b\right) \mu =-2; \sigma =\frac{1}{2} \\ \left(c\right) \mu =0; \sigma =2 \end{gathered}$$

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 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.

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. 

A variable has a density curve whose equation is$y=2x$ for $0<x<1$ and $y=0$ and otherwisse,

(a) Graph the denisty curve of this variable.

(b) Show that the area under this density curve to the left of any number x between 0 and 1 equals $x^2$.

What percentage of all possible observations of the variable 

(c) lie between $\frac{1}{2}$ and $\frac{3}{4}$.

(d) are atleast $\frac{1}{4}$.

A variable has a density curve whose equation is $y=1$ for $0<x<1$ and $y=0$ and otherwisse,

(a) Graph the density curve of this variable.

(b) Show that the area under this density curve to the left of any number x between 0 and 1 equals x.

What percentage of all possible observations of the variable 

(c) lie between $\frac{1}{4}$ and $\frac{3}{4}$.

(d) exceed $\frac{7}{8}$.

Waiting for the Train. A commuter train arrives punctually at a station every half hour. Each morning, a commuter named John leaves his house and casually strolls to the train station. The time, in minutes, that John waits for a train is a variable with density curve $y=\frac{1}{30} for 0<x<30, y=0$and otherwise.

(a) Graph the density curve of this variable.

(b) Show that the area under this density curve to the left of any number x between 0 and 30 equals $\frac{x}{30}$.

What percentage of all possible observations of the variable 

(c) less than 5 minutes.

(d) between 10 and 15 minutes.

(e) atleast 20 minutes.

A petri dish is a small, shallow dish of thin glass or plastic, used especially for cultures in bacteriology. A 2-inch-radius petri dish, containing nutrients upon which bacteria can multiply, is smeared with a uniform suspension of bacteria. Subsequently, spots indicating colonies of bacteria appear. The distance (in inches) of the center of the first spot to appear from the center of the petri dish is a variable with density curve $y=\frac{x}{2}$ for 0<x<2, and y=0 otherwise

 Fire Loss. The loss, in millions of dollars, due to a fire in a commercial building is a variable with density curve$y=1-x/2$ for $0<x<2$ and $y=0$ otherwise. Using the fact that the area of a triangle equals one-half its base times its height, we find that the area under this density curve to the left of any number $x$ between $0 and 2$equals $x-x^2/4$

a. Graph the density curve of this variable.

b. What percentage of losses exceed $1.5$ million?

 Female College Students. Refer to Example 6.3 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 6.3 on page 256 .

a. The area under the 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 actual percentage of female students who are shorter than 61 inches.

c. Compare your answers from parts (a) and (b).

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 $206\mathrm{mg}/\mathrm{dL}$ (milligrams per deciliter) and a standard deviation of $44.7\mathrm{mg}/\mathrm{dL}$. Let $x$ denote serum 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 sketch the distribution of $z$.

d. The percentage of U.S. females $20$ years old or older who have a serum total cholesterol level between $150\mathrm{mg}/\mathrm{dL}$ and $250\mathrm{mg}/\mathrm{dL}$ is equal to the area under the standard normal curve between-------- .

New York City $10$-km Run. As reported in Rumner's World magazine, the times of the finishers in the New York City $10$-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 ---- .

e. The percentage of finishers with times less than $75$ minutes is equal to the area under the standard normal curve that lies to the ----- of ----- .

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

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 normallydistributed with a mean of $18.14$mm and a standard deviation of $1.76$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 $16$mm and $17$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 exceeding $19$mm is equal to the area under the standard normal curve that lies 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 $10$-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 _____ .

e. The percentage of finishers with times less than $75$ minutes is equal to the area under the standard normal curve that lies to the _____ of _____ .

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 $52.0$g and standard deviation $17.2$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 $50$ g and $60$g is equal to the area under he standard normal curve between ____ and ____ .

e. The percentage of adult green sea urchins with weights above $40$g 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 $44.7$ mg/dL. Let $x$ denote serum total cholesterol level for U.S. females $20$ years old or older.

a. Sketch the distribution of the variable $x$.

b. Obtain the standardization version,$z$ of $x$.

c. Identify and sketch 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 $250$ mg/dL is equal to the area under the standard normal curve between _____ .

6.31 Waiting for the Train. A commuter train arrives punctually at a station every half hour. Each morning, a commuter named John leaves his house and casually strolls to the train station. The time, in minutes, that John waits for the train is a variable with density curve $y=1/30$ for $0<x<30$, and $y=0$ otherwise.
a. Graph the density curve of this variable.
b. Show that the area under this density curve to the left of any number $x$ between 0 and 30 equals $x/30$.

What percentage of the time does John wait for the train
c. less than 5 minutes?
d. between 10 and 15 minutes?
e. at least 20 minutes?

6.32 Bacteria on a Petri Dish. A petri dish is a small, shallow dish of thin glass or plastic, used especially for cultures in bacteriology. A 2 -inch-radius petri dish, containing nutrients upon which bacteria can multiply, is smeared with a uniform suspension of bacteria. Subsequently. spots indicating colonies of bacteria appear. The distance of the center of the first spot to appear from the center of the petri dish is a variable with density curve $y=x/2$ for $0<x<2$, and $y=0$ otherwise.
a. Graph the density curve of this variable.
b. Show that the area under this density curve to the left of any number $x$between 0 and 2 equals $x^2/4$.
What percentage of the time is the distance of the center of the first spot to appear from the center of the petri dish
c. at most 1 inch?
d. between 0.25 inch and 1.5 inches?
e. more than 0.75 inch?

Desert Samaritan Hospital in Mesa, Arizona keeps record of its emergency-room traffic. Beginning at \(6:00\)PM on any given day, the elapsed time in hours until the first patient arrives is a variable with density curve \(y=6.9e^{-6.9x}\) for \(x>0\) and \(y=0\) otherwise. Here \(e\) is Euler's number which is approximately \(2.71828\). Most calculators have an \(e-\)key. Using calculus, it can be shown that the area under this density curve to the left of any number \(x\) greater than \(0\) equals \(1-e^{-6.9x}\).

a. Graph the density curve of this variable.

b. What percentage of the time does the first patient arrive between \(6:15\)PM and \(6:30\)PM?

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

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.

 In the paper "Cloudiness: Note on a Novel Case of Frequency" (Procedings of the Royul Sociery 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.

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.

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 Techical Journal. 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.

a. Construct a relative-frequency histogram of these wrong-number 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.

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 site. 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.

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 site. 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.

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). pp. 10-14]

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.

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.

Ages of Mothers. From the document National Vital Statistics Reports, a publication of the National Centre for Health Statistics, we obtained the following frequency distribution for the ages of women who became mothers during one year.

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. 

6.49 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 T7-83/84 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? Explain your answer.
f. Obtain a histogram of the 1500 observations, and compare your result to your expectation in part (e).

Refer to the simulation of human gestation periods discussed in Example \(6.4\) on page \(259\).

a. Sketch the normal curve for human gestation periods

b. Simulate \(1000\) human gestation periods.

c. Approximately what values would you expect for the sample mean and sample 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 \(1000\) observation to look like? Explain your answer.

f. Obtain a histogram of the \(1000\) observations, and compare your result to your expectation in part (e). 

Without consulting Table II, explain why the area under the standard normal curve that lies to the right of $0$ is $0.5$.

According to Table II, the area under the standard normal curve that lies to the left of $-2.08$ is $0.0188$. Without further consulting Table II, determine the area under the standard normal curve that lies to the right of $2.08$. Explain your reasoning.

According to Table II, the area under the standard normal curve that lies to the left of $0.43$ is $0.6664$. Without further consulting Table II, determine the area under the standard normal curve that lies to the right of $0.43$. Explain your reasoning.

According to Table II, the area under the standard normal curve that lies to the left of $1.96$ is $0.975$. Without further consulting Table II, determine the area under the standard normal curve that lies to the left of $-1.96$. Explain your reasoning.

Property $4$ of Key Fact $6.5$ states that most of the area under the standard normal curve lies between $-3$ and $3$ . Use Table II to determine precisely the percentage of the area under the standard normal curve that lies between $-3$ and $3$.

The area under the standard normal curve that lies to the left of a $z$-score is always strictly between ------ and  ------.

Explain how Table II is used to determine the area under the standard normal curve that lies

a. to the left of a specified $z$-score.

b. to the right of a specified $z$-score.

c. between two specified $z$-scores.

With which normal distribution is the standard normal curve associated?

Use Table to obtain the areas under the standard normal curve. Sketch a standard normal curve and shade the area of interest in each problem.

Find the area under the standard normal curve that lies to the right of

a. $2.02$.

b. $-0.56$.

c. $-4$.

Use Table to obtain the areas under the standard normal curve. Sketch a standard normal curve and shade the area of interest in each problem.

Find the area under the standard normal curve that lies

a. either to the left of $-2.12$ or to the right of $1.67$.

b. either to the left of $0.63$ or to the right of $1.54$.

Use Table to obtain the areas under the standard normal curve. Sketch a standard normal curve and shade the area of interest in each problem.

Find the area under the standard normal curve that lies

a. either to the left of $-1$ or to the right of $2$.

b. either to the left of $-2.51$ or to the right of $-1$.

In each part, find the area under the standard normal curve that lies between the specified $z$-scores, sketch a standard normal curve and shade the area of interest.

a. $-1$ and $1$

b. $-2$ and $2$

c. $-3$ and $3$

Why is the standard normal curve sometimes referred to as the z-curve?

Use Table II to obtain the areas under the standard normal curve required in Exercises 6.59-6.66. Sketch a standard normal curve and shade the area of interest in each problem.

Determine the area under the standard normal curve that lies to the left of

a. $2.24$

b. $-1.56$

c. $0$

d. $-4$