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

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.

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 16 gas stations is

• A $\overline{X}$ ~ $N(4.59,0.10)$
• B $\overline{X}$ ~ $N(4.59,\frac{0.10}{\sqrt{16}})$
• C $\overline{X}$ ~ $N(4.59,\frac{0.10}{16})$
• D $\overline{X}$ ~ $N(4.59,\frac{16}{0.10})$

What is the probability that the average price for 16 gas stations is over $4.69?

• 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 $4.55.

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

For the Charter School Problem (Example 6) in Central Limit Theorem: Using the Central Limit Theorem , calculate the following using the normal approximation to the binomial.

• A Find the probability that less than 100 favor a charter school for grades K - 5.
• B Find the probability that 170 or more favor a charter school for grades K - 5.
• C Find the probability that no more than 140 favor a charter school for grades K - 5.
• D Find the probability that there are fewer than 130 that favor a charter school for grades K - 5.
• E Find the probability that exactly 150 favor a charter school for grades K - 5.

If you either have access to an appropriate calculator or computer software, try calculating these probabilities using the technology. Try also using the suggestion that is at the bottom of Central Limit Theorem: Using the Central Limit Theorem for finding a website that calculates binomial probabilities.

Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for 96 days. Use the normal approximation to the binomial to calculate the following probabilities. Round the standard deviation to 4 decimal places.

• A Find the probability that Janice is the driver at most 20 days.
• B Find the probability that Roberta is the driver more than 16 days.
• C Find the probability that Barbara drives exactly 24 of those 96 days.

If you either have access to an appropriate calculator or computer software, try calculating these probabilities using the technology. Try also using the suggestion that is at the bottom of Central Limit Theorem: Using the Central Limit Theorem for finding a website that calculates binomial probabilities.