6.10 Homework  (Page 2/2)

In the 1992 presidential election, Alaska’s 40 election districts averaged 1956.8 votes per district for President Clinton. The standard deviation was 572.3. (There are only 40 election districts in Alaska.) The distribution of the votes per district for President Clinton was bell-shaped. Let $X=$ number of votes for President Clinton for an election district. (Source: The World Almanac and Book of Facts )

• a State the approximate distribution of $X$ . $X$ ~
• b Is 1956.8 a population mean or a sample mean? How do you know?
• c Find the probability that a randomly selected district had fewer than 1600 votes for President Clinton. Sketch the graph and write the probability statement.
• d Find the probability that a randomly selected district had between 1800 and 2000 votes for President Clinton.
• e Find the third quartile for votes for President Clinton.

Suppose that the duration of a particular type of criminal trial is known to be normally distributed with a mean of 21 days and a standard deviation of 7 days.

• a In words, define the random variable $X$ . $X=$
• b $X$ ~
• c If one of the trials is randomly chosen, find the probability that it lasted at least 24 days. Sketch the graph and write the probability statement.
• d 60% of all of these types of trials are completed within how many days?

Terri Vogel, an amateur motorcycle racer, averages 129.71 seconds per 2.5 mile lap (in a 7 lap race) with a standard deviation of 2.28 seconds . The distribution of her race times is normally distributed. We are interested in one of her randomly selected laps. (Source: log book of Terri Vogel)

• a In words, define the random variable $X$ . $X=$
• b $X$ ~
• c Find the percent of her laps that are completed in less than 130 seconds.
• d The fastest 3% of her laps are under _______ .
• e The middle 80% of her laps are from _______ seconds to _______ seconds.

Thuy Dau, Ngoc Bui, Sam Su, and Lan Voung conducted a survey as to how long customers at Lucky claimed to wait in the checkout line until their turn. Let $X=$ time in line. Below are the ordered real data (in minutes):

• a Calculate the sample mean and the sample standard deviation.
• b Construct a histogram. Start the $x-\text{axis}$ at $-0\text{.}\text{375}$ and make bar widths of 2 minutes.
• c Draw a smooth curve through the midpoints of the tops of the bars.
• d In words, describe the shape of your histogram and smooth curve.
• e Let the sample mean approximate $\mu$ and the sample standard deviation approximate $\sigma$ . The distribution of $X$ can then be approximated by $X$ ~
• f Use the distribution in (e) to calculate the probability that a person will wait fewer than 6.1 minutes.
• g Determine the cumulative relative frequency for waiting less than 6.1 minutes.
• h Why aren’t the answers to (f) and (g) exactly the same?
• i Why are the answers to (f) and (g) as close as they are?
• j If only 10 customers were surveyed instead of 50, do you think the answers to (f) and (g) would have been closer together or farther apart? Explain your conclusion.

Suppose that Ricardo and Anita attend different colleges. Ricardo’s GPA is the same as the average GPA at his school. Anita’s GPA is 0.70 standard deviations above her school average. In complete sentences, explain why each of the following statements may be false.

• a Ricardo’s actual GPA is lower than Anita’s actual GPA.
• b Ricardo is not passing since his z-score is zero.
• c Anita is in the 70th percentile of students at her college.

Below is a sample of the maximum capacity (maximum number of spectators) of sports stadiums. The table does not include horse racing or motor racing stadiums. (Source: http://en.wikipedia.org/wiki/List_of_stadiums_by_capacity )

• a Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums (the data).
• b Construct a histogram of the data.
• c Draw a smooth curve through the midpoints of the tops of the bars of the histogram.
• d In words, describe the shape of your histogram and smooth curve.
• e Let the sample mean approximate $\mu$ and the sample standard deviation approximate $\sigma$ . The distribution of $X$ can then be approximated by $X$ ~
• f Use the distribution in (e) to calculate the probability that the maximum capacity of sports stadiums is less than 67,000 spectators.
• g Determine the cumulative relative frequency that the maximum capacity of sports stadiums is less than 67,000 spectators. Hint: Order the data and count the sports stadiums that have a maximum capacity less than 67,000. Divide by the total number of sports stadiums in the sample.
• h Why aren’t the answers to (f) and (g) exactly the same?

What is the median recovery time?

• A 2.7
• B 5.3
• C 7.4
• D 2.1

What is the z-score for a patient who takes 10 days to recover?

• A 1.5
• B 0.2
• C 2.2
• D 7.3

What is the probability of spending more than 2 days in recovery?

• A 0.0580
• B 0.8447
• C 0.0553
• D 0.9420

The 90th percentile for recovery times is?

• A 8.89
• B 7.07
• C 7.99
• D 4.32

Based upon the above information and numerically justified, would you be surprised if it took less than 1 minute to find a parking space?

• A Yes
• B No
• C Unable to determine

Find the probability that it takes at least 8 minutes to find a parking space.

• A 0.0001
• B 0.9270
• C 0.1862
• D 0.0668

Seventy percent of the time, it takes more than how many minutes to find a parking space?

• A 1.24
• B 2.41
• C 3.95
• D 6.05

If the mean is significantly greater than the standard deviation, which of the following statements is true?

• I The data cannot follow the uniform distribution.
• II The data cannot follow the exponential distribution..
• III The data cannot follow the normal distribution.

• A I only
• B II only
• C III only
• D I, II, and III