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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?

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-------- .

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.

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.

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.

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

a. $-0.87.$

b.$3.56$

c. $5.12$.

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

a. $-1.07$

b. $0.6$

c. $0$

d.$4.2$

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 $1.54$.

6.68 Use Table II to obtain each shaded area under the standard normal curve.

Determine$z_{0.33}$

Determine$Z_{0.015}$