7.3 Using the Central Limit Theorem

In the United States, a robbery occurs every two minutes, on average, according to a number of studies. 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 robberies in the United States.
2. Find the median, the first quartile, and the third quartile for the sum of sample times of robberies i…

A study was done regarding attendance at Broadway shows in New York City. The age range of the attendees was 14 to 61. The mean age was 30.9 years with a standard deviation of nine years.

1. In a sample of 25 attendees, what is the probability that the mean age 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 sample of 49 at…

Use the following information to answer the next ten 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 is the mean and standard deivat…

Use the following information to answer the next ten 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 ten 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 ten 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 < 2,450).

Use the following information to answer the next ten 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 seven 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 IQR for the mean amount of time 64 batteries last.

Use the following information to answer the next eight 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 eight 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 eight 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. 8.625
2. 30.25
3. 27.625
4. 46.75
5. 32.875
6. 18.25
7. 5
8. 0.125
9. 2.9375
10. 6.875
11. 28.25
12. 24.25
13. 21
14. 1.5
15. 30.25
16. 71
17. 43.5
18. 49.25
19. 2.5625
20. 31
21. 16.5
22. 9.5
23. 18.5
24. 18
25. 9
26. 10.5
27. 16.625
28. 1.25
29. 18
30. 12.87
31. 7
32. 12.875
33. 2.875
34. 60.25
35. 29.25

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…

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 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.

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 teachers in a particular elementary school district are norm…

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

M&M candies large candy bags 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 he listed weights in the table came from randomly selected candies…

For each problem, wherever possible, provide graphs and use the 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 ran…

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

Certain coins have an average weight of 5.201 grams 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?