# Confidence Interval, Margin of Error & Sample size

Chapter 7 - Section 2

Problem 20

Finding Margin of Error. Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level.

90% confidence; the sample size is 1780, of which 35% are successes.

Problem 30

Gender Selection - The Genetics and IVF Institute conducted a clinical trial of the YSORT method designed to increase the probability of conceiving a boy. As of this writing, 152 babies were born to parents using the YSORT method, and 127 of them were boys.

a. What is the best point estimate of the population proportion of boys born to parents using the YSORT method?

b. Use the sample data to construct a 99% confidence interval estimate of the percentage of the boys born to parents using the YSORT method.

c. Based on the results, does the YSORT method appear to be effective? Why or why not?

Problem 42

Determining Sample Size - Find the minimum sample size required to estimate a population proportion or percentage.

Cell Phones - As the newly hired manager of a company that provides cell phone service, you want to determine the percentage of adults in your state who live in a household with cell phones and no land-line phones. How many adults must you survey? Assume that you want to be 90% confident that the sample percentage is within four percentage points of the true population percentage.

a. Assume that nothing is known about the percentage of adults who live in a household with cell phones and no land-line phones.

b. Assume that a recent survey suggests that about 8% of adults live in a household with cell phones and no land-line phones (based on data from the National Health Interview Survey).

Chapter 7 - Section 3

Problem 14

Finding Sample Size - Use the given information to find the minimum sample size required to estimate an unknown population mean.

Braking Distances - How many cars must be randomly selected and tested in order to estimate the mean braking distance of registered cars in the United States. We want 99% confidence that the sample mean is within 2 ft of the population mean, and the population standard deviation is known to be 7 ft.

Problem 26

Birth Weights - A simple random sample of birth weights in the United States has a mean of 3433 g. The standard deviation of all birth weights is 495 g.

a. Using a sample size of 75, construct a 95% confidence interval estimate of the mean birth weight in the United States.

b. Using a sample size of 75,000, construct a 95% confidence interval estimate of the mean birth weight in the United States.

c. Which of the preceding confidence intervals is wider? Why?

Problem 34

Finding Sample Size - Find the indicated sample size.

Grade Point Average - A researcher wants to estimate the mean grade point average of all current college students in the U.S. She has developed a procedure to standardize scores from colleges using something other than a scale between 0 and 4. How many grade point averages must be obtained so that the sample mean is within 0.1 of the population mean? Assume that a 90% confidence level is desired. Also assume that a pilot study showed that the population standard deviation is estimated to be 0.88.

Chapter 7 - Section 4

Problem 18

Constructing Confidence Intervals. Construct the confidence interval.

Birth Weights - A random sample of the birth weights of 186 babies has a mean of 3103 g and a standard deviation of 696 g (based on data from "Cognitive Outcomes of Preschool Children with Prenatal Cocaine Exposure," by Singer et al., Journal of the American Medical Association, Vol. 291, No. 20). These babies were born to mothers who did not use cocaine during their pregnancies.

a. What is the best point estimate of the mean weight of babies born to mothers who did not use cocaine during their pregnancies?

b. Construct a 95% confidence interval estimate of the mean birth weight for all such babies.

c. Compare the confidence interval from part (b) to this confidence interval obtained from birth weights of babies born to mothers who used cocaine during pregnancy: 2608 g < mean < 2792 g. Does cocaine use appear to affect the birth weight of a baby?

Problem 30

Constructing Confidence Intervals. Construct the confidence interval.

Ages of Presidents - Listed bewlo are the ages of the Presidents of the U.S. at the times of their inaugurations. Construct a 99% confidence interval estimate of the mean age of presidents at the tiems of their inaugurations. What is the population? Does the confidence interval provide a good estimate of the population mean? Why or why not?

42 43 46 46 47 48 49 49 50 51 51 51 51 51 52 52 54 54 54 54 54 55

55 55 55 56 56 56 57 57 57 57 58 60 61 61 61 62 64 64 65 68 69

Chapter 7 - Section 5

Problem 16

Determining Sample Size - Assume that each sample is a simple random sample obtained from a normally distributed population. Use Table 7-2 on page 376 to find the indicated sample size.

Find the minimum sample size needed to be 95% confident that the sample variance is within 20% of the population variance.

Table 7-2

Sample Size for Standard Deviaition 2 Sample size for Standard Deviation

To be 95% of the value of SD 2 To be 95% of the value of SD,

confident that the sample size n confident that the sample size n

s2 is within should be at least s is within should be at least

1% 77,208 1% 19,205

5% 3,149 5% 768

10% 806 10% 192

20% 211 20% 48

30% 98 30% 21

40% 57 40% 12

50% 38 50% 8

Sample Size for Standard Deviaition 2 Sample size for Standard Deviation

To be 95% of the value of SD 2 To be 95% of the value of SD,

confident that the sample size n confident that the sample size n

s2 is within should be at least s is within should be at least

1% 133,449 1% 33,218

5% 5,458 5% 1,336

10% 1,402 10% 336

20% 369 20% 85

30% 172 30% 38

40% 101 40% 22

50% 68 50% 14

Problem 24

Finding Confidence Intervals - Assume that each sample is a simple random sample obtained from a population with a normal distribution.

a. Comparing Waiting Lines - The listed values are waiting times (in minutes) of customers at the Jefferson Valley Bank, where customers enter a single waiting line that feeds three teller windows. Construct a 95% confidence intervalf or the population standard deviation.

6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7

b. The listed values are waiting times (in minutes) of customers at the Bank of Providence, where customers may enter any one of three different lines that have formed at three teller windows. Construct a 95% confidence interval for the population standard deviation.

4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0

c. Interpret the results found in parts (a) and (b). Do the confidence intervals suggest a difference in the variation among waiting times? Which arrangement seems better: the single-line system or the multiple-line system?

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#### Solution Summary

The solution provides step by step method for the calculation of confidence interval, margin of error and sample size. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

Statistics 1: 10 problems of confidence intervals, sample sizes, distribution

You work for a consumer advocate agency and want to find the mean repair cost of a washing machine. As part of your study, you randomly select 45 repair costs and find the mean to be $117.00. The standard deviation is $18.90. Complete parts (a) and (b).

a. Construct a 90% confidence interval for the population mean repair costs.

b. Change the sample size to n = 90. Construct a 90% confidence interval for the population mean repair cost.

Which confidence interval is wider? Explain. Choose the correct answer below.

2. People were polled on how many books they read the previous year. How many subjects are needed to estimate the number of books read the previous year within one book with 95% confidence? Initial survey results indicate that σ = 10.7 books

3. Repair Costs: Microwaves In a random sample of five microwave ovens, the mean repair cost was $80.00 and the standard deviation was $13.00. Assume the variable is normally distributed and use a t-distribution to construct a 90% confidence interval for the population mean µ. What is the margin of error of µ?

4. A researcher wishes to estimate, with 99% confidence, the proportion of adults who have high-speed internet access. Her estimate must be accurate within 5% of the true proportion.

a) Find the minimum sample size needed, using a prior study that found that 38% of the respondents said they have high-speed internet access.

b) No preliminary estimate is available. Find the minimum sample size needed.

5. An election poll reported that a candidate had an approval rating of 31% with a margin of error E of 2%. Construct a confidence interval for the proportion of adults who approve of the candidate.

6. Find the margin of error for the given values of c, s and n.

c = 0.95, s = 3.1, n = 36

7. Construct the 98% confidence interval for the population mean µ.

c = 0.98, x-bar = 16.2, s = 8.0 and n = 70

8. Construct the indicated confidence interval for the population mean µ using (a) a t-distribution. (b) If you had incorrectly used a normal distribution, which interval would be wider?

c = 0.99, x-bar = 14.2, s = 2.0, n = 9

9. Construct the indicated confidence interval for the population mean µ using a t-distribution.

c = 0.80, x-bar = 108, s = 10, n = 13

10. A researcher wishes to estimate, with 90% confidence, the percentage of adults who support abolishing the penny. His estimate must be accurate within 5% of the true proportion.

a) Find the minimum sample size needed, using a prior study that found that 38% of the respondents said they support abolishing the penny.

See attached file.

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