Project #88264 - Statistics and Probability

1.   The heights of eighteen-year-old men are approximately normally distributed with a mean (µ) of 68 inches and a standard deviation (σ) of 3 inches. What is the probability that an eighteen-year-old man selected at random is taller than 70 inches?

2.   Suppose that, instead of randomly selecting one 18-year-old male, you randomly select ten 18-year-old males. What is the probability that the average height for these 18-year-old males is more than 70 inches?

3.   The average age for employees at an amusement park is 24 years old with a standard deviation of 2.5 years. Suppose random samples of 40 employees are selected. What would the distribution of average ages from samples of this size look like? Why?

4.   Use the above amusement park scenario to answer the following questions:

a.   What would the average be for all sample means from samples of this size?

b.   What would the standard error be for all sample means from samples of this size?

5.     What is the probability if you randomly selected 40 employees and averaged their ages together that the sample mean would be between 23 and 25 years?

6.   What makes the normal distribution a probability distribution?

7.   In a normal distribution, what percentage of the area under a normal curve is between µ - σ and µ + σ?

8.   In a normal distribution, what percentage of the area under a normal curve is between µ - σ and µ + 2σ?

9.   Suppose you are interested in how long it takes to get your food at a restaurant. Now, suppose this distribution is approximately normal with an average of eight minutes and a standard deviation of two minutes. If you made a control chart for this data, what would be the highest control limit?

10.                 Using the above scenario, suppose someone gets the food after exactly eleven minutes. How many standard deviations from the mean is this value of eleven?

11.                 Using the above scenario, what is the probability that someone would get the food after more than eleven minutes?

12.                 Using the standard normal distribution, find the area below z=-0.8.

13.                 Using the standard normal distribution, find the area above z=-0.8.

14. Using the standard normal distribution, find the area between z=-0.8 and z=2.1.

 

  1. What percentage of the area under the normal curve lies

 

a.    To the left of the mean?

b.    Between μ – 2σ and  μ + 2σ?

c.     More than three standard deviations above the mean?

 

16. Over an entire summer, an amusement park gets an average of 21.7 people per day that have to go to the infirmary. Some days it is higher than this. Some days it is lower than this. The standard deviation is 4.2. The distribution for the amount of people treated is approximately normal.

 

a.    For a ten day period, here are the number of people treated each of those ten days:

 

Day

1

2

3

4

5

6

7

8

9

10

Number treated

25

19

17

15

20

24

30

19

16

23

 

Make a control chart for the daily number treated, and plot this data on that chart. Do the data indicate that the number of people treated is “in control”? Explain your answer.

 

b.    For a ten day period, here are the number of people treated each of those 10 days:

 

Day

1

2

3

4

5

6

7

8

9

10

Number treated

20

15

12

21

24

28

32

36

35

37

 

Make a control chart for the daily number treated, and plot this data on that chart. Do the data indicate that the number of people treated is “in control”? Explain your answer.

 

  1. A normal distribution has mean = 12 and standard deviation = 3.

 

a.    The z-score corresponding to x = 18.

b.    Find the raw score corresponding to z = -1.5.

 

18.John received an 85 percent on a history test and a 78 percent on a Spanish test. For the history test, the class mean was 82 percent and standard deviation 10. For the Spanish test, the class mean was 74 percent and standard deviation 2. On which test did he do better relative to the rest of the class?

 

19.Find the specified areas under the standard normal curve:

 

a.    To the left of z = .56

b.    To the right of z = 1.3

c.     To the right of z = -2.2

d.    Between z = -1.2 and z = 2.1

e.    P(-1.78 < z < -1.23)

 

20.If the mean of a normal distribution is 40 and the standard deviation is 4, find P(38 < x < 46).

 

21.Find z such that...

a.    10% of the standard normal curve lies to the right of z.

b.    90% of the standard normal curve lies between -z and z.

 

 

22. A local band is going on a U.S. Summer tour, and they average about 2000 people per concert, with a standard deviation of about 400. Assume that these concert numbers follow a normal distribution.

a.    If a concert is selected at random, what is the probability that there were more than 2500 people at that concert?

b.    If a concert is selected at random, what is the probability that there were less than 1800 people at that concert?

c.     If a concert is selected at random, what is the probability that there were between 1800 and 2500 people at that concert?

d.    For a concert to be in the top 10% as far as attendance, at least how many people would need to attend the concert?

 

23. Define what a sample statistic is. Give three examples from your everyday life of sample statistics.

 

24. Define what a sampling distribution is. Using one of your three examples from the previous problem, explain a possible sampling distribution from that example.

 

25. Define what the standard error of a sample distribution is.

 

26. The heights of 18 year old females are approximately normally distributed with a mean of 64 inches and a standard deviation of 3 inches.

 

    1. What is the probability that an 18-year-old woman selected at random is between 63 and 65 inches tall?
    2. Suppose samples of 25 18-year-old females are taken at a time. Describe the sampling distribution of the sample mean and compute the mean and standard deviation of this sampling distribution.
    3. Find the z-score corresponding to a sample mean of 66 inches for a sample of 25 females.
    4. Find the probability that a sample mean from a sample like this would be higher than 66 inches.
    5. Based on the probability found in the previous part, would a sample like this be unusual?
    6. If a random sample of 25 18-year-old females is selected, what is the probability that the mean height for this sample is between 63 and 65 inches?

 

27.                 Suppose you are trying to estimate the average miles per gallon for a new brand of car. You take a random sample of 40 cars, and, for this sample, the average miles per gallon is 32 and the standard deviation for this sample is 2.2. Answer the following questions in a Word document:

28.                 What is the population mean you are trying to estimate?

29.                 What value is the point estimate for the population mean? Describe, in your own words, what the point estimate represents.

30.                 Suppose you would like to construct a 90 percent confidence interval. What would be the margin of error for this interval?

31.                 What would be the lower and upper limits for this 90 percent confidence interval?

32. Now that you have constructed the interval, define its meaning in your own words.

33.Answer true or false for each part, and if false, explain your answer.

a.    The point estimate for the population mean, µ, of an x distribution is x-bar, computed from a random sample of the x distribution.

b.    Every random sample of the same size from a given population will produce exactly the same confidence interval for µ.

c.     For the same random sample, when the confidence level is reduced, the confidence interval for µ becomes wider.

 

34.Use tables or software to find the t-value for a 90% confidence interval when the sample size is 20.

 

35.A random sample of size 64 has sample mean 24 and sample standard deviation 4.

d.    Is it appropriate to use the t distribution to compute a confidence interval for the population mean? Why or why not?

e.    Construct a 95% confidence interval for the population mean.

f.     Explain the meaning of the confidence interval you just constructed.

 

36.How much do adult male grizzly bears weigh in the wild? Six adult males were captured, tagged and released in California and here are their weights:

         480, 580, 470, 510, 390, 550

g.    What is the point estimate for the population mean?

h.    Construct at 90% confidence interval for the population average weight of all adult male grizzly bears in the wild.

i.      Interpret the confidence interval in the context of this problem.

 

37.After going to a fast food restaurant, customers are asked to take a survey. Out of a random sample of 340 customers, 290 said their experience was “satisfactory.” Let p represent the proportion of all customers who would say their experience was “satisfactory.”

j.      What is the point estimate for p?

k.    Construct a 99% confidence interval for p.

l.      Give a brief interpretation of this interval.

 

38.In your own words, define each of the following terms that are used in hypothesis testing:

a.    The null hypothesis.

b.    The alternative hypothesis.

c.     The test statistic

d.    The p-value

e.    Type I Error

f.     Type II Error

 

39.                  Suppose the p-value for a right-tailed test is .0245.

a.    What would be your conclusion at the .05 level of significance?

b.    What would the p-value have been if it were a two-tailed test?

 

40.A random sample has 42 values. The sample mean is 9.5 and the sample standard deviation is 1.5. Use a level of significance of 0.02 to conduct a left-tailed test of the claim that the population mean is 10.0.

a.    Are the requirements met to run a test like this?

b.    What are the hypotheses for this test?

c.     Compute the test statistic and the p-value for this test.

d.    What is your conclusion at the 0.02 level of significance?

 

41.MTV states that 75% of all college students have seen at least one episode of their TV show “Jersey Shore”. Last month, a random sample of 120 college students was selected and asked if they had seen at least one episode of the show. Out of the 120, 85 of them said they had seen at least one episode. Is there enough evidence to claim the population proportion of all college student that have watched at least one episode is less than 75% at the 0.05 level of significance?

a.    Are the requirements met to run a test like this?

b.    What are the hypotheses for this test?

c.     Compute the test statistic and the p-value for this test.

d.    What is your conclusion at the 0.05 level of significance?

 

Subject Mathematics
Due By (Pacific Time) 10/22/2015 12:00 am
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