6.7 Homework  (Page 3/4)

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 considered wedge shaped (many very poor people, very few middle income people, and few to many wealthy people). Suppose we pick a country with a wedge distribution. Let the average salary be $2000 per year with a standard deviation of $8000. We randomly survey 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?

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. (Source: U.S. Dept. of Agriculture)

• a In words, $X=$
• b In words, $\overline{X}=$
• c $\overline{X}\text{~}$
• d The IQR for $\overline{X}$ is from _______ acres to _______ acres.

The stock closing prices of 35 U.S. semiconductor manufacturers are given below. (Source: Wall Street Journal )

• 8.625
• 30.25
• 27.625
• 46.75
• 32.875
• 18.25
• 5
• 0.125
• 2.9375
• 6.875
• 28.25
• 24.25
• 21
• 1.5
• 30.25
• 71
• 43.5
• 49.25
• 2.5625
• 31
• 16.5
• 9.5
• 18.5
• 18
• 9
• 10.5
• 16.625
• 1.25
• 18
• 12.875
• 7
• 12.875
• 2.875
• 60.25
• 29.25

• a In words, $X=$
• b
  • i $\overline{x}=$
  • ii $s_x=$
  • iii $n=$
• c Construct a histogram of the distribution of the averages. Start at $x=-0\text{.}\text{0005}$ . Make bar widths of 10.
• d In words, describe the distribution of stock prices.
• e Randomly average 5 stock prices together. (Use a random number generator.) Continue averaging 5 pieces together until you have 10 averages. List those 10 averages.
• f Use the 10 averages from (e) to calculate:
  • i $\overline{x}=$
  • ii $\overline{s_x}=$
• g Construct a histogram of the distribution of the averages. Start at $x=-0\text{.}\text{0005}$ . Make bar widths of 10.
• h Does this histogram look like the graph in (c)?
• i In 1 - 2 complete sentences, explain why the graphs either look the same or look different?
• j Based upon the theory of the Central Limit Theorem, $\overline{X}\text{~}$