8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

2:23 minutes

Problem 7.1.33a

Textbook Question

E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?

a. Assume that nothing is known about the rate of e-cigarette usage among adults.

Verified step by step guidance

Step 1: Identify the formula for determining the required sample size for estimating a population proportion. The formula is: n = (Z^2 * p * (1 - p)) / E^2, where n is the required sample size, Z is the z-score corresponding to the desired confidence level, p is the estimated population proportion, and E is the margin of error.

Step 2: Determine the values for the variables in the formula. Since the confidence level is 95%, the corresponding z-score (Z) is approximately 1.96. The margin of error (E) is given as 1.5%, which should be converted to a decimal: E = 0.015. Since nothing is known about the rate of e-cigarette usage, we use the most conservative estimate for p, which is 0.5 (this maximizes the product p * (1 - p)).

Step 3: Substitute the values into the formula. Replace Z with 1.96, p with 0.5, and E with 0.015 in the formula: n = (1.96^2 * 0.5 * (1 - 0.5)) / 0.015^2.

Step 4: Simplify the numerator of the formula. Calculate 1.96^2, then multiply it by 0.5 and (1 - 0.5). This will give you the value of the numerator.

Step 5: Simplify the denominator of the formula. Square the margin of error (0.015^2), then divide the numerator by the denominator to find the required sample size (n). Round up to the nearest whole number, as sample size must be an integer.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed in a study to achieve a desired level of precision. It takes into account the confidence level, margin of error, and variability in the population. In this case, we need to calculate how many adults should be surveyed to estimate the current e-cigarette usage rate with a specified confidence level and margin of error.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within the confidence interval. A 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 95 of those intervals would contain the true population parameter. This concept is crucial for understanding the reliability of the survey results.

Margin of Error

The margin of error indicates the range within which the true population parameter is expected to fall, based on the sample data. A margin of error of 1.5 percentage points means that the estimate of e-cigarette usage could be 1.5 percentage points higher or lower than the sample result. This concept is essential for assessing the precision of the survey findings and ensuring that the results are statistically valid.

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Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?

b. Assume that 11% of consumers have a smartphone and plan to upgrade to a new model.

Textbook Question

c. Given that the required sample size is relatively small, could you simply survey the people that you know?

Textbook Question

Women Who Give Birth An epidemiologist plans to conduct a survey to estimate the percentage of women who give birth. How many women must be surveyed in order to be 99% confident that the estimated percentage is in error by no more than two percentage points?

c. What is wrong with surveying randomly selected adult women?

Textbook Question

One-Sided Confidence Interval A one-sided claim about a population proportion is a claim that the proportion is less than (or greater than) some specific value. Such a claim can be formally addressed using a one-sided confidence interval for p, which can be expressed as p<p+E or p>p-E, where the margin of error E is modified by replacing za/2 with za. (Instead of dividing between two tails of the standard normal distribution, put all of it in one tail.) The Chapter Problem refers to a Sallie Mae survey of 950 undergraduate students, and 53% of the survey subjects take online courses. Use that data to construct a one-sided 95% confidence interval that would be suitable for helping to determine whether the percentage of all undergraduates who take online courses is greater than 50%.

Textbook Question

No Failures According to the Rule of Three, when we have a sample size n with x=0 successes, we have 95% confidence that the true population proportion has an upper bound of 3/n. (See “A Look at the Rule of Three,” by Jovanovic and Levy, American Statistician, Vol. 51, No. 2.)

a. If n independent trials result in no successes, why can’t we find confidence interval limits by using the methods described in this section?

Textbook Question

b. In a study of failure rates of computer hard drives, 45 Toshiba model MD04ABA500V hard drives were tested and there were no failures. What is the 95% upper bound for the percentage of failures for the population of all such hard drives?

Textbook Question

Seating Choice In a 3M Privacy Filters poll, respondents were asked to identify their favorite seat when they fly, and the results include these responses: window, window, other, other. Letting “window” and letting “other”, those four responses can be represented as {1, 1, 0, 0}. Here are ten bootstrap samples for those responses: [Image]

Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the proportion of respondents who indicated their favorite seat is “window.”

Textbook Question

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Use the bootstrap method to construct a 95% confidence interval estimate of the proportion of lawsuits that are dropped or dismissed. Use 1000 bootstrap samples. How does the result compare to the confidence interval found in Exercise 16 “Medical Malpractice” from Section 7-1?

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