7.1 Homework: clt (modified r. bloom)  (Page 3/3)

Note: Parts d,e,f of this problem have been changed from original version of textbook.

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 Find the probability that an individual teacher earns more than $40,000.
• e Find the probability that the average salary for the sample is more than $40,000.
• f Find the probability that the average salary for the sample is more than $50,000.
• g Find the 90th percentile for an individual teacher’s salary.
• h Find the 90th percentile for the average teachers’ salary for samples of 10 teachers.
• i If we surveyed 70 teachers instead of 10, graphically, how would that change the distribution for $\overline{X}$ ?
• j If each of the teachers in this elementary school district 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 Question removed from text
• e Question removed from 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?

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 )

• 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, $\overline{X}\text{~}$

Use the Initial Public Offering data (see “Table of Contents) to do this problem.

• a In words, $X=$
• b
  • i ${\mu }_X=$
  • ii ${\sigma }_X=$
  • iii $n=$
• c Construct a histogram of the distribution. Start at $x=-0\text{.}\text{50}$ . Make bar widths of $5.
• 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 15 averages. List those 15 averages.
• f Use the 15 averages from (e) to calculate the following:
  • i $\overline{x}=$
  • ii $\overline{s_x}=$
• g Construct a histogram of the distribution of the averages. Start at $x=-0\text{.}\text{50}$ . Make bar widths of $5.
• h Does this histogram look like the graph in (c)? Explain any differences.
• 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, $\overline{X}\text{~}$

The 90th percentile sample average wait time (in minutes) for a sample of 100 riders is:

• A 315.0
• B 40.3
• C 38.5
• D 65.2

Would you be surprised, based upon numerical calculations, if the sample average wait time (in minutes) for 100 riders was less than 30 minutes?

• A Yes
• B No
• C There is not enough information.

Which of the following is NOT TRUE about the distribution for averages?

• A The mean, median and mode are equal
• B The area under the curve is one
• C The curve never touches the x-axis
• D The curve is skewed to the right

The distribution to use for the average cost of gasoline for the 30 gas stations is

• A $\overline{X}$ ~ $N(2.59,0.10)$
• B $\overline{X}$ ~ $N(2.59,\frac{0.10}{\sqrt{30}})$
• C $\overline{X}$ ~ $N(2.59,\frac{0.10}{30})$
• D $\overline{X}$ ~ $N(2.59,\frac{30}{0.10})$

What is the probability that the average price for 30 gas stations is over $2.69?

• A Almost zero
• B 0.1587
• C 0.0943
• D Unknown

Find the probability that the average price for 30 gas stations is less than $2.55.

• A 0.6554
• B 0.3446
• C 0.0142
• D 0.9858
• E 0