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In the 1992 presidential election, Alaska’s 40 election districts averaged 1,956.8 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.

a. State the approximate distribution of X.

b. Is 1,956.8 a population mean or a sample mean? How do you know?

c. Find the probability that a randomly selected district had fewer than 1,600 votes for President Clinton. Sketch the graph and write the probability statement.

d. Find the probability that a randomly selected district had between 1,800 and 2,000 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 seven days.

a. In words, define the random variable 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. Sixty percent of all trials of this type are completed within how many days?

What does a z-score measure?

Terri Vogel, an amateur motorcycle racer, averages $129.71$seconds per $2.5$mile lap (in a seven-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.

a. In words, define the random variable $X$.

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. Table below displays the ordered real data (in minutes):

a. Calculate the sample mean and the sample standard deviation.

b. Construct a histogram.

c. Draw a smooth curve through the midpoints of the tops of the bars.

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 because his $z$-score is zero.

c. Anita is in the ${70}^{\text{th}}$ percentile of students at her college.

Table 6.4 shows a sample of the maximum capacity (maximum number of spectators) of sports stadiums. The table does not include horse-racing or motor-racing stadiums.

a. Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums (the data).

b. Construct a histogram.

c. Draw a smooth curve through the midpoints of the tops of the bars of the histogram.

An expert witness for a paternity lawsuit testifies that the length of a pregnancy is normally distributed with a mean of $280$days and a standard deviation of $13$days. An alleged father was out of the country from $240$to $306$days before the birth of the child, so the pregnancy would have been less than $240$days or more than $306$days long if he was the father. The birth was uncomplicated, and the child needed no medical intervention. What is the probability that he was NOT the father? What is the probability that he could be the father? Calculate the localid="1653472319552" $z$-scores first, and then use those to calculate the probability.

A NUMMI assembly line, which has been operating since $1984$, has built an average of $6,000$ cars and trucks a week. Generally, $10\%$ of the cars were defective coming off the assembly line. Suppose we draw a random sample of $n=100$ cars. Let $X$ represent the number of defective cars in the sample. What can we say about $X$ in regard to the $68-95-99.7$ empirical rule (one standard deviation, two standard deviations and three standard deviations from the mean are being referred to)? Assume a normal distribution for the defective cars in the sample.

We flip a coin \(100\) times \((n = 100)\) and note that it only comes up heads \(20% (p = 0.20)\) of the time. The mean and standard deviation for the number of times the coin lands on heads is \(\mu=20\) and \(\sigma=4\) (verify the mean and standard deviation). Solve the following:

a. There is about a \(68%\) chance that the number of heads will be somewhere between ___ and ___.

b. There is about a ____chance that the number of heads will be somewhere between \(12\) and \(28\).

c. There is about a ____ chance that the number of heads will be somewhere between eight and \(32\).