7.10 Central limit theorem: homework  (Page 2/3)

Suppose that a category of world class 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.

• a $\overline{X}\text{~}$
• 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.

Suppose that the length of research papers is uniformly distributed from 10 to 25 pages. We survey a class in which 55 research papers were turned in to a professor. The 55 research papers are considered a random collection of all papers. We are interested in the average length of the research papers.

• a In words, $X=$
• b $X\text{~}$
• c ${\mu }_X=$
• d ${\sigma }_X=$
• e In words, $\overline{X}=$
• f $\overline{X}\text{~}$
• g In words, $\mathrm{\Sigma X}=$
• h $\mathrm{\Sigma X}\text{~}$
• i Without doing any calculations, do you think that it’s likely that the professor will need to read a total of more than 1050 pages? Why?
• j Calculate the probability that the professor will need to read a total of more than 1050 pages.
• k Why is it so unlikely that the average length of the papers will be less than 12 pages?

The length of songs in a collector’s CD collection is uniformly distributed from 2 to 3.5 minutes. Suppose we randomly pick 5 CDs from the collection. There is a total of 43 songs on the 5 CDs.

• a In words, $X=$
• b $X\text{~}$
• c In words, $\overline{X}\text{=}$
• d $\overline{X}\text{~}$
• e Find the first quartile for the average song length.
• f The IQR (interquartile range) for the average song length is from _______ to _______.

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6500. We randomly survey 10 teachers from that district.

• a In words, $X=$
• b In words, $\overline{X}=$
• c $\overline{X}\text{~}$
• d In words, $\mathrm{\Sigma X}=$
• e $\mathrm{\Sigma X}\text{~}$
• f Find the probability that the teachers earn a total of over $400,000.
• g Find the 90th percentile for an individual teacher’s salary.
• h Find the 90th percentile for the average teachers’ salary.
• i If we surveyed 70 teachers instead of 10, graphically, how would that change the distribution for $\overline{X}$ ?
• j If each of the 70 teachers received a $3000 raise, graphically, how would that change the distribution for $\overline{X}$ ?

The distribution of income in some Third World countries is not considered normal (many very poor people, very few middle income people, and few to many wealthy people). We pick a country consistent with this non-normal distribution. We find the average salary be $2000 per year with a standard deviation of $8000. We randomly surveyed 1000 residents of that country.

• a In words, $X=$
• b In words, $\overline{X}=$
• c $\overline{X}\text{~}$
• 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 1000 residents will be from $2000 to $2100 than from $2100 to $2200?

The average length of a maternity stay in a U.S. hospital is said to be 2.4 days with a standard deviation of 0.9 days. We randomly survey 80 women who recently bore children in a U.S. hospital.

• a In words, $X=$
• b In words, $\overline{X}=$
• c $\overline{X}\text{~}$
• d In words, $\mathrm{\Sigma X}=$
• e $\mathrm{\Sigma X}\text{~}$
• f Is it likely that an individual stayed more than 5 days in the hospital? Why or why not?
• g Is it likely that the average stay for the 80 women was more than 5 days? Why or why not?
• h Which is more likely:
  • i an individual stayed more than 5 days; or
  • ii the average stay of 80 women was more than 5 days?
• i If we were to sum up the women’s stays, is it likely that, collectively they spent more than a year in the hospital? Why or why not?