6.7 Normal distribution: homework

According to a study done by De Anza students, the height for Asian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Asian adult male is randomly chosen. Let $X=$ height of the individual.

• a $X$ ~ $\text{\_\_\_\_\_\_\_}\text{(\_\_\_\_\_\_\_,\_\_\_\_\_\_\_)}$
• b Find the probability that the person is between 65 and 69 inches. Include a sketch of the graph and write a probability statement.
• c Would you expect to meet many Asian adult males over 72 inches? Explain why or why not, and justify your answer numerically.
• d The middle 40% of heights fall between what two values? Sketch the graph and write the probability statement.

IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let $X=$ IQ of an individual.

• a $X$ ~ $\text{\_\_\_\_\_\_\_}\text{(\_\_\_\_\_\_\_,\_\_\_\_\_\_\_)}$
• b Find the probability that the person has an IQ greater than 120. Include a sketch of the graph and write a probability statement.
• c Mensa is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the Mensa organization. Sketch the graph and write the probability statement.
• d The middle 50% of IQs fall between what two values? Sketch the graph and write the probability statement.

The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of 10. Suppose that one individual is randomly chosen. Let $X=$ percent of fat calories.

• a $X$ ~ $\text{\_\_\_\_\_\_\_}\text{(\_\_\_\_\_\_\_,\_\_\_\_\_\_\_)}$
• b Find the probability that the percent of fat calories a person consumes is more than 40. Graph the situation. Shade in the area to be determined.
• c Find the maximum number for the lower quarter of percent of fat calories. Sketch the graph and write the probability statement.

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet.

• a If $X=$ distance in feet for a fly ball, then $X$ ~ $\text{\_\_\_\_\_\_\_}\text{(\_\_\_\_\_\_\_,\_\_\_\_\_\_\_)}$
• b If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.
• c Find the 80th percentile of the distribution of fly balls. Sketch the graph and write the probability statement.

In China, 4-year-olds average 3 hours a day unsupervised. Most of the unsupervised children live in rural areas, considered safe. Suppose that the standard deviation is 1.5 hours and the amount of time spent alone is normally distributed. We randomly survey one Chinese 4-year-old living in a rural area. We are interested in the amount of time the child spends alone per day. (Source: San Jose Mercury News )

• a In words, define the random variable $X$ . $X=$
• b $X$ ~
• c Find the probability that the child spends less than 1 hour per day unsupervised. Sketch the graph and write the probability statement.
• d What percent of the children spend over 10 hours per day unsupervised?
• e 70% of the children spend at least how long per day unsupervised?