49
Previously, De Anza statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $0.88. Suppose that we randomly pick 25 daytime statistics students.
 In words, \(Χ\) = ____________
 \(Χ \sim\) _____(_____,_____)
 In words, \(\overline X\) = ____________
 \(\overline X \sim\) ______ (______, ______)
 Find the probability that an individual had between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.
 Find the probability that the average of the 25 students was between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.
 Explain why there is a difference in part e and part f.
 Answer

 \(Χ\) = amount of change students carry
 \(Χ \sim E(0.88, 0.88)\)
 \(\overline X\) = average amount of change carried by a sample of 25 students.
 \(\overline X \sim N(0.88, 0.176)\)
 \(0.0819\)
 \(0.1882\)
 The distributions are different. Part 1 is exponential and part 2 is normal.
50.
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.
 If \(\overline X\) = average distance in feet for 49 fly balls, then \(\overline X \sim\) _______(_______,_______)
 What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for \(\overline X\). Shade the region corresponding to the probability. Find the probability.
 Find the 80th percentile of the distribution of the average of 49 fly balls.
51.
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.
 In words, \(Χ =\) _____________
 In words, \(\overline X\) = _____________
 \(\overline X \sim\) _____(_____,_____)
 Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.
 Would you be surprised if one taxpayer finished his or her Form 1040 in more than 12 hours? In a complete sentence, explain why.
52.
Suppose that a category of worldclass runners are 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 \(\overline X\) the average of the 49 races.
 \(\overline X \sim\) _____(_____,_____)
 Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.
 Find the \(80^{th}\) percentile for the average of these 49 marathons.
 Find the median of the average running times.
53.
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.
 In words, \(Χ\) = _________
 \(Χ \sim\) _____________
 In words, \(\overline X\) = _____________
 \(\overline X \sim\) _____(_____,_____)
 Find the first quartile for the average song length.
 The \(IQR\) (interquartile range) for the average song length is from _______–_______.
54.
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.
 In words, \(Χ\) = _____________
 In words, \(\overline X\) = _____________
 \(\overline X \sim\) _____(_____,_____)
 The \(IQR\) for \(\overline X\) is from _______ acres to _______ acres.
55.
Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.
 When the sample size is large, the mean of \(\overline X\) is approximately equal to the mean of \(Χ\).
 When the sample size is large, \(\overline X\) is approximately normally distributed.
 When the sample size is large, the standard deviation of \(\overline X\) is approximately the same as the standard deviation of \(Χ\).
56.
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 \(\overline X\) = average percent of fat calories.
 \(\overline X \sim\) ______(______, ______)
 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.
 Find the first quartile for the average percent of fat calories.
57.
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 $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.
 In words, \(Χ\) = _____________
 In words, \(\overline X\) = _____________
 \(\overline X \sim\) _____(_____,_____)
 How is it possible for the standard deviation to be greater than the average?
 Why is it more likely that the average of the 1,000 residents will be from $2,000 to $2,100 than from $2,100 to $2,200?
58.
Which of the following is NOT TRUE about the distribution for averages?
 The mean, median, and mode are equal.
 The area under the curve is one.
 The curve never touches the xaxis.
 The curve is skewed to the right.
59.
The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. The distribution to use for the average cost of gasoline for the 16 gas stations is:
a. \(\overline X \sim N(4.59, 0.10)\)
b. \(\overline X \sim N\left(4.59, \frac{0.10}{\sqrt{16}}\right)\)
c. \(\overline X \sim N\left(4.59, \frac{16}{0.10}\right)\)
d. \(\overline X \sim N\left(4.59, \frac{\sqrt{16}}{0.10}\right)\)