7.3 Using the Central Limit Theorem

Use the information in the example, but use a sample size of 55 to answer the following questions.

1. Find P($\bar{x}$ < 7).
2. Find P(Σx > 170).
3. Find the 80^th percentile for the mean of 55 scores.
4. Find the 85^th percentile for the sum of 55 scores.

Use the information in the example, but change the sample size to 144.

1. Find P(20 < $\bar{x}$ < 30).
2. Find P(Σx is at least 3000).
3. Find the 75^th percentile for the sample mean excess time of 144 customers.
4. Find the 85^th percentile for the sum of 144 excess times used by customers.

U.S. scientists studying a certain medical condition discovered that a new person is diagnosed every two minutes, on average. Suppose the standard deviation is 0.5 minutes and the sample size is 100.

1. Find the median, the first quartile, and the third quartile for the sample mean time of diagnosis in the United States.
2. Find the median, the first quartile, and the third quartile for the sum of sampl…

Based on data from the National Health Survey, women between the ages of 18 and 24 have an average systolic blood pressures (in mm Hg) of 114.8 with a standard deviation of 13.1. Systolic blood pressure for women between the ages of 18 to 24 follows a normal distribution.

1. If one woman from this population is randomly selected, find the probability that her systolic blood pressure is greater than 12…

A study was done about a medical condition that affects a certain group of people. The age range of the people was 14–61. The mean age was 30.9 years with a standard deviation of nine years.

1. In a sample of 25 people, what is the probability that the mean age of the people is less than 35?
2. Is it likely that the mean age of the sample group could be more than 50 years? Interpret the results.
3. In a sa…

According to data from an aerospace company, the 757 airliner carries 200 passengers and has doors with a mean height of 72 inches. Assume for a certain population of men we have a mean of 69 inches inches and a standard deviation of 2.8 inches.

1. What mean doorway height would allow 95 percent of men to enter the aircraft without bending?
2. Assume that half of the 200 passengers are men. What mean doo…

In a city, 46 percent of the population favors 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.

Use the following information to answer the next 10 exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.

1. What is the distribution for the weights of one 25-pound lifting weight? What are the mean and standard deiva…

Use the following information to answer the next 10 exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.

Find the probability that the mean actual weight for the 100 weights is greater than 25.2.

Use the following information to answer the next 10 exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.

Find the 90^th percentile for the mean weight for the 100 weights.

Use the following information to answer the next 10 exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.

1. What is the distribution for the sum of the weights of 100 25-pound lifting weights?
2. Find P(Σx < 2450).

Use the following information to answer the next 10 exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.

Find the 90^th percentile for the total weight of the 100 weights.

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

1. What is the standard deviation?
2. What is the parameter m?

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

What is the distribution for the mean length of time 64 batteries last?

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

Find the probability that the sample mean is between 7 and 11.

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

Find the interquartile range (IQR) for the mean amount of time 64 batteries last.

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.

Find P(Σx > 420).

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.

Find the 15^th percentile for the sums.

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.

Find the third quartile for the sums.

The closing stock prices of 35 U.S. semiconductor manufacturers are given as follows:

1. In words, Χ = ______________.
2. 1. $\bar{x}$ = _____
   2. s_x = _____
   3. n = _____
3. Construct a histogram of the distribution of the averages. Start at x = –0.0005. Use bar widths of 10.
4. In words, describe the distribution of the stock prices.
5. Randomly average five stock prices together. (Use a random number generator.) Continue ave…

Use the following information to answer the next three exercises: Richard’s Furniture Company delivers furniture from 10 a.m. to 2 p.m. continuously and uniformly. We are interested in how long (in hours) past the 10 a.m. start time that individuals wait for their delivery.

The average wait time is:

1. one hour
2. two hours
3. two and a half hours
4. four hours

Use the following information to answer the next two exercises: The time to wait for a particular rural bus is distributed uniformly from zero to 75 minutes. One hundred riders are randomly sampled to learn how long they waited.

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

1. 315.0
2. 40.3
3. 38.5
4. 65.2

Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.

What's the approximate probability that the average price for 16 gas stat…

Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.

Suppose in a local kindergarten through 12^th grade (K–12) school district…

Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.

X ~ N(60, 9). Suppose that you form random samples of 25 from this distri…

Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.

Salaries for managers in a restaurant chain are normally distributed with…

For each problem, wherever possible, provide graphs and use a calculator.

NeverReady batteries has engineered a newer, longer-lasting AAA battery. The company claims this battery has an average life span of 17 hours with a standard deviation of 0.8 hours. Your statistics class questions this claim. As a class, you randomly select 30 batteries and find that the sample mean life span is 16.7 hours.…

For each problem, wherever possible, provide graphs and use a calculator.

Large bags of a brand of multicolored candies have a claimed net weight of 396.9 g. The standard deviation for the weight of the individual candies is 0.017 g. The following table is from a stats experiment conducted by a statistics class.

The bag contained 465 candies and the listed weights in the table came from randomly…

For each problem, wherever possible, provide graphs and use a calculator.

Your company has a contract to perform preventive maintenance on thousands of air conditioners in a large city. Based on service records from previous years, the time that a technician spends servicing a unit averages one hour with a standard deviation of one hour. In the coming week, your company will service a simple rando…

For each problem, wherever possible, provide graphs and use a calculator.

Certain coins have an average weight of 5.201 g with a standard deviation of 0.065 g. If a vending machine is designed to accept coins whose weights range from 5.111 g to 5.291 g, what is the expected number of rejected coins when 280 randomly selected coins are inserted into the machine?