Chapter 7: The Central Limit Theorem

According to Boeing data, the $757$airliner carries $200$passengers and has doors with a height of $72$inches. Assume for a certain population of men we have a mean height of $69.0$inches and a standard deviation of $2.8$ inches.
a. What doorway height would allow $95\%$of men to enter the aircraft without bending?
b. Assume that half of the $200$passengers are men. What mean doorway height satisfies the condition that there is a$0.95$probability that this height is greater than the mean height of $100$men?

c. For engineers designing the $757$ , which result is more relevant: the height from part a or part b? Why?

According to Boeing data, the $757$airliner carries$200$passengers and has doors with a height of $72$inches. Assume for a certain population of men we have a mean height of $69.0$inches and a standard deviation of $2.8$inches.

a. What doorway height would allow $95\%$of men to enter the aircraft without bending?

b. Assume that half of the $200$passengers are men. What mean doorway height satisfies the condition that there is a $0.95$probability that this height is greater than the mean height of$100$men?

c. For engineers designing the $757$, which result is more relevant: the height from part a or part b? Why?

In a city, $46$percent of the population favor the incumbent, Dawn Morgan, for mayor. A simple random sample of$500$is taken. Using the continuity correction factor, find the probability that at least $250$ favor Dawn Morgan for mayor.

72. Which of the following is NOT TRUE about the theoretical distribution of sums?

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 length of time taken on the SAT for a group of students is normally distributed with a mean of 2.5 hours and a standard deviation of 0.25 hours. A sample size of n = 60 is drawn randomly from the population. Find the probability that the sample mean is between two hours and three hours.

The mean number of minutes for app engagement by a table use is $8.2$ minutes. Suppose the standard deviation is one minute. Take a sample size of $70$.

a. What is the probability that the sum of the sample is between seven hours and ten hours? What does this mean in context of the problem?

b. Find the $84th$and $16th$percentiles for the sum of the sample. Interpret these values in context.

Suppose that the duration of a particular type of criminal trial is known to have a mean of $21$days and a standard deviation of seven days. We randomly sample nine trials.

a. In words,$\Sigma X=\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

b.$\Sigma X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

c. Find the probability that the total length of the nine trials is at least $225$days.

d. Ninety percent of the total of nine of these types of trials will last at least how long?

In an article on Flurry Blog, a gaming marketing gap for men between the ages of 30 and 40 is identified. You are researching a startup game targeted at the 35-year-old demographic. Your idea is to develop a strategy game that can be played by men from their late 20s through their late 30s. Based on the article’s data, industry research shows that the average strategy player is 28 years old with a standard deviation of 4.8 years. You take a sample of 100 randomly selected gamers. If your target market is 29- to 35-year-olds, should you continue with your development strategy?

74. Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from two to six pounds with a mean of four pounds and a standard deviation of 1.1547. We randomly survey 64 homes with children.

a. In words, X=

b. The distribution is

c. In words, $\sum x$=

d. $\sum x~$

e. Find the probability that the total weight of open boxes is less than 250 pounds.

f. Find the$35th$percentile for the total weight of open boxes of cereal.

Cans of a cola beverage claim to contain 16 ounces. The amounts in a sample are measured and the statistics are $n=34,\overline{x}=16.01$ounces. If the cans are filled so that $\mu =16.00$ounces (as labeled) and $\sigma =0.143$ounces, find the probability that a sample of 34 cans will have an average amount greater than $16.01$ounces. Do the results suggest that cans are filled with an amount greater than 16 ounces?