91Ó°ÊÓ

Yoonie is a personnel manager in a large corporation. Each month she must review $16$of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of $1.2$hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let $\stackrel{-}{x}$be the random variable representing the meantime to complete the $16$reviews. Assume that the $16$reviews represent a random set of reviews.

Find the $95$th percentile for the meantime to complete one month's reviews. Sketch the graph.

a.

b. The $95$th Percentile =____________

Previously, De Anza statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $\backslash (0.88$. Suppose that we randomly pick $25$daytime statistics students.

a. In words,$\mathrm{Î\S }=\_\_\_\_\_\_\_\_\_\_\_\_$

b.$\mathrm{Î\S }~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

c.role="math" localid="1651578876947" $Inwords,\overline{X}=\_\_\_\_\_\_\_\_\_\_\_\_$

d. $\overline{X}~\_\_\_\_\_\_(\_\_\_\_\_\_,\_\_\_\_\_\_)$

e. Find the probability that an individual had between $\backslash )0.80and\backslash (1.00$. Graph the situation, and shade in the area to be determined.

f. Find the probability that the average of the 25 students was between $\backslash )0.80and\$1.00$. Graph the situation, and shade in the area to be determined.

g. Explain why there is a difference in part e and part f.

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. We randomly sample 49 fly balls.

a. If $X$the average distance in feet for 49 fly balls, then $\stackrel{¯}{X}~$

b. What is the probability that the 49 balls traveled an average of fewer than 240 feet? Sketch the graph. Scale the horizontal axis for$\stackrel{¯}{X}$. Shade the region corresponding to the probability. Find the probability.

c. Find the $80th$percentile of the distribution of the average of 49 fly balls.

According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form $1040$is $10.53$hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is two hours. Suppose we randomly sample $36$taxpayers.

a. In words, $\mathrm{Î\S }=\_\_\_\_\_\_\_\_\_\_\_\_\_$

b. In words,$\overline{X}=\_\_\_\_\_\_\_\_\_\_\_\_\_$

c. $X¯~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

d. Would you be surprised if the $36$taxpayers finished their Form $1040$s in an average of more than $12$hours? Explain why or why not in complete sentences.

e. Would you be surprised if one taxpayer finished his or her Form $1040$in more than $12$hours? In a complete sentence, explain why.

64. Suppose that a category of world-class runners is known to run a marathon ( 26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races. Let $\stackrel{-}{X}$be the average of the 49 races.

a. role="math" localid="1652281768411" $\stackrel{-}{X}~$(___)

b. Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.

c. Find the 80th percentile for the average of these 49 marathons.

d. Find the median of the average running times.

The length of songs in a collector’s iTunes album collection is uniformly distributed from two to $3.5$minutes. Suppose we randomly pick five albums from the collection. There are a total of $43$songs on the five albums.

a. In words,$\mathrm{Î\S }=\_\_\_\_\_\_\_\_\_$

b.$\mathrm{Î\S }~\_\_\_\_\_\_\_\_\_\_\_\_\_$

c. In words,$\overline{X}=\_\_\_\_\_\_\_\_\_\_\_\_\_$

d.$\overline{X}~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

e. Find the first quartile for the average song length, $\overline{X}$.

f. The IQR (interquartile range) for the average song length, $\overline{X}$, is from ___ - ___.

66. In 1940 the average size of a U.S. farm was 174 acres. Let's say that the standard deviation was 55 acres. Suppose we randomly survey 38 farmers from 1940.

a. In words, X=

b. In words, $\stackrel{-}{X}$=

c. $\stackrel{-}{X~}$

d. The IQR for $\stackrel{-}{X}$is from acres to acres.

Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.

a. When the sample size is large, the mean of $\stackrel{¯}{X}$is approximately equal to the mean of $X$.

b. When the sample size is large, $\stackrel{¯}{X}$is approximately normally distributed.

c. When the sample size is large, the standard deviation of $\stackrel{¯}{X}$is approximately the same as the standard deviation of $X$.

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 about ten. Suppose that $16$individuals are randomly chosen. Let role="math" localid="1648361500255" $\stackrel{¯}{X}=$average percent of fat calories.

a. $\stackrel{¯}{X}~$_____ (______, ______)

b. For the group of $16$, find the probability that the average percent of fat calories consumed is more than five. Graph the situation and shade in the area to be determined.

c. Find the first quartile for the average percent of fat calories.

The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $\backslash (2,000$per year with a standard deviation of $\backslash )8,000$. We randomly survey $1,000$residents of that country.

a. In words,$\mathrm{Î\S }=\_\_\_\_\_\_\_\_\_\_\_\_\_$

b. In words,$\overline{X}=\_\_\_\_\_\_\_\_\_\_\_\_\_$

c.$X¯~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

d. How is it possible for the standard deviation to be greater than the average?

e. Why is it more likely that the average of the $1,000$residents will be from $\backslash (2,000$to $\backslash )2,100$than from $\backslash (2,100to\backslash )2,200$?