### Find the area under the standard normal curve

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##### Reference no: EM13280532

Part -1:

1. If the random variable z is the standard normal score and P(z < a) < 0.5, then a > 0. Why or why not?

2. Given a binomial distribution with n = 20 and p = 0.76, would the normal distribution provide a reasonable approximation? Why or why not?

3. Find the area under the standard normal curve for the following:

(A) P(z < -0.74)
(B) P(-0.87 < z < 0)
(C) P(-2.03 < z < 1.66)

4. Find the value of z such that approximately 10.26% of the distribution lies between it and the mean.

5. Assume that the average annual salary for a worker in the United States is \$31,000 and that the annual salaries for Americans are normally distributed with a standard deviation equal to \$7,500. Find the following:

(A)What percentage of Americans earn below \$20,000?

(B)What percentage of Americans earn above \$45,000?

6. X has a normal distribution with a mean of 80.0 and a standard deviation of 3.5. Find the following probabilities:
(A) P(x < 75.0)
(B) P(78.0 < x < 83.0)

(C) P(x > 85.0)

7. Answer the following:

(A) Find the binomial probability P(x = 4), where n = 12 and p = 0.70.

(B) Set up, without solving, the binomial probability P(x is at most 4) using probability notation.

(C) How would you find the normal approximation to the binomial probability P(x = 4) in part A? Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations.

Part -2:

1. Consider a population with and .

(A) Calculate the z-score for from a sample of size 12.
(B) Could this z-score be used in calculating probabilities using Table 3 in Appendix B of the text? Why or why not?

2. Given a level of confidence of 99% and a population standard deviation of 15, answer the following:
(A) What other information is necessary to find the sample size (n)?
(B) Find the Maximum Error of Estimate (E) if n = 93. Show all work.

3. A sample of 75 golfers showed that their average score on a particular golf course was 91.32 with a standard deviation of 6.85.
Answer each of the following (show all work
and state the final answer to at least two decimal places.):
(A) Find the 95% confidence interval of the mean score for all 75 golfers.
(B) Find the 95% confidence interval of the mean score for all golfers if this is a sample of 120 golfers instead of a sample of 75.
(C) Which confidence interval is larger and why?

4. Assume that the population of heights of male college students is approximately normally distributed with mean μ of 70.07 inches and standard deviation σ of 6.48 inches. A random sample of 93 heights is obtained. Show all work.
(A) Find the mean and standard error of the distribution
(B) Find

5. The diameters of grapefruits in a certain orchard are normally distributed with a mean of 5.11 inches and a standard deviation of 0.49 inches. Show all work.
(A) What percentage of the grapefruits in this orchard is larger than 5.18 inches?
(B) A random sample of 100 grapefruits is gathered and the mean diameter is calculated. What is the probability that the sample mean is greater than 5.18 inches?

6. A researcher is interested in estimating the noise levels in decibels at area urban hospitals. She wants to be 95% confident that her estimate is correct. If the standard deviation is 4.02, how large a sample is needed to get the desired information to be accurate within 0.58 decibels? Show all work.

Part -3

1. Consider a normal population with µ = 25 and σ = 7.0.
(A) Calculate the standard score for a value x of 23.
(B) Calculate the standard score for a randomly selected sample of 45 with = 23.
(C) Explain why the standard scores of 23 are different between A and B above.

2. Assume that the mean SAT score in Mathematics for 11th graders across the nation is 500, and that the standard deviation is 100 points. Find the probability that the mean SAT score for a randomly selected group of 150 11th graders is between 475 and 525.

3. Assume that a sample is drawn and z(α/2) = 1.65 and σ = 15. Answer the following questions:

(A) If the Maximum Error of Estimate is 0.05 for this sample, what would be the sample size?

(B) Given that the sample Size is 400 with this same z(α/2) and σ, what would be the Maximum Error of Estimate?

(C) What happens to the Maximum Error of Estimate as the sample size gets larger?

(D) What effect does the answer to C above have to the size of the confidence interval?

4. By measuring the amount of time it takes a component of a product to move from one workstation to the next, an engineer has estimated that the standard deviation is 3.22 seconds.

Answer each of the following (show all work):

(A) How many measurements should be made in order to be 98% certain that the maximum error of estimation will not exceed 0.5 seconds?

(B) What sample size is required for a maximum error of 1.5 seconds?

5. A 90% confidence interval estimate for a population mean was computed to be (27.6, 47.2). Determine the mean of the sample, which was used to determine the interval estimate (show all work).

6. A study was conducted to estimate the mean amount spent on birthday gifts for a typical family having two children. A sample of 140 was taken, and the mean amount spent was \$176.85. Assuming a standard deviation equal to \$36.73, find the 98% confidence interval for μ, the mean for all such families (show all work).

7. A confidence interval estimate for the population mean is given to be (42.43, 51.02). If the standard deviation is 17.845 and the sample size is 47, answer each of the following (show all work):

(A) Determine the maximum error of the estimate, E.

(B) Determine the confidence level used for the given confidence interval.

Part -4:

1. (A) Classify the following as an example of nominal, ordinal, interval, or ratio level of measurement, and state why it represents this level: eye color
(B) Determine if this data is qualitative or quantitative: weight
(C) In your own line of work, give one example of a discrete and one example of a continuous random variable, and describe why each is continuous or discrete.

2. A construction company ordered a variety of lumber grades for their new housing developments for the purpose of examining characteristics of durability and longevity of their construction projects. The company recorded durability and longevity data on each development over a period of five years. At the end of the five years, they reported the results of their research to all construction organizations through a national publication.
I. What is the population?
II. What is the sample?
III. Is the study observational or experimental? Justify your answer.
IV. What are the variables?
V. For each of those variables, what level of measurement (nominal, ordinal, interval, or ratio) was used to obtain data from these variables?

3. Construct both an ungrouped and a grouped frequency distribution for the data given below:
92 93 86 81 85 81 86 88 89 93
83 92 81 93 92 85 93 82 95 86

4. Given the following frequency distribution, find the mean, variance, and standard deviation. Please show all of your work.

Class Frequency
51-53 7
54-56 19
57-59 9
60-62 13
63-65 15

5. The following data lists the average monthly snowfall for January in 15 cities around the US:

13 12 24 44 32 28 12 17
39 47 11 33 45 17 22

Find the mean, variance, and standard deviation. Please show all of your work.

6. Rank the following data in increasing order and find the positions and values of both the 27th percentile and 67th
percentile. Please show all of your work.
2 7 3 7 9 3 1 8 0 3 5 9

7. For the table that follows, answer the following questions:
x y
1 -8
2 -11
3 -14
4

- Would the correlation between x and y in the table above be positive or negative?

- Find the missing value of y in the table.

- How would the values of this table be interpreted in terms of linear regression?

- If a "line of best fit" is placed among these points plotted on a coordinate system, would the slope of this line be positive or negative?

8. Determine whether each of the distributions given below represents a probability distribution. Justify your answer.

(A)
x 1 2 3 4
P(x) 1/5 9/25 2/5 1/25

(B)
x 3 6 8
P(x) 0.7 1/12 0.22

(C)
x 20 30 40 50
P(x) 13/25 0.25 -0.01 6/25

9. A set of 50 data values has a mean of 31 and a variance of 25.

I. Find the standard score (z) for a data value = 35.
II. Find the probability of a data value > 35.
III. Find the probability of a data value < 35.
Show all work.

10. Answer the following:
(A) Find the binomial probability P(x = 4), where n = 12 and p = 0.30.
(B) Set up, without solving, the binomial probability P(x is at most 4) using probability notation.
(C) How would you find the normal approximation to the binomial probability P(x = 4) in part A? Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations.

11. Assume that the population of heights of female college students is approximately normally distributed with mean μ of 64.43 inches and standard deviation σ of 4.61 inches. A random sample of 89 heights is obtained. Show all work.
(A) Find
(B) Find the mean and standard error of the distribution
(C) Find
(D) Why is the formula required to solve (A) different than (C)?

12. Determine the critical region and critical values for z that would be used to test the null hypothesis at the given level of significance, as described in each of the following:

(A) and , α = 0.01
(B) and , α = 0.10
(C) and , α = 0.05

13. Describe what a type I and type II error would be for each of the following null hypotheses:
: There is no way to pass this driving test.

14. A researcher claims that the average age of people who buy theatre tickets is 51. A sample of 30 is selected and their ages are recorded as shown below. The standard deviation is 5. At α
= 0.01 is there enough evidence to reject the researcher's claim? Show all work.
70 47 34 45 71 54 52 52 32 56
45 46 60 54 48 54 47 46 58 47
61 45 72 56 46 50 80 55 47 54

15. Write a correct null and alternative hypothesis for testing the claim that the mean life of a battery for a cell phone is at least 75 hours.

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