8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

2:30 minutes

Problem 7.1.37a

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?

a. Assume that nothing is known about the worldwide percentage of consumers who intend to buy the new model.

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 90%, the corresponding z-score (Z) is approximately 1.645. The margin of error (E) is given as 3%, or 0.03. Since nothing is known about the population proportion, assume p = 0.5, which maximizes the required sample size.

Step 3: Substitute the values into the formula. Replace Z with 1.645, p with 0.5, and E with 0.03 in the formula: n = (1.645^2 * 0.5 * (1 - 0.5)) / 0.03^2.

Step 4: Simplify the numerator of the formula. Calculate 1.645^2, then multiply it by 0.5 and (1 - 0.5).

Step 5: Simplify the denominator of the formula. Calculate 0.03^2, then divide the simplified numerator by the simplified denominator to find the required sample size, n.

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Key Concepts

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

Sample Size Determination

Sample size determination is a statistical method used to calculate the number of observations or replicates needed in a survey to achieve a desired level of confidence and precision. In this case, the marketing team needs to determine how many consumers to survey to estimate the percentage of potential buyers accurately, ensuring that the estimate is within three percentage points of the true population percentage.

Confidence Level

The confidence level represents the degree of certainty that the sample accurately reflects the population. A 90% confidence level means that if the survey were conducted multiple times, 90% of the time the results would fall within the specified margin of error. This concept is crucial for understanding how reliable the survey results will be in predicting consumer behavior.

Margin of Error

The margin of error indicates the range within which the true population parameter is expected to lie, based on the sample results. In this scenario, a margin of error of three percentage points means that the estimated percentage of consumers intending to purchase the new iPhone could vary by plus or minus three percentage points from the actual percentage. This concept is essential for assessing the precision of the survey results.

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Textbook Question

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a. Assume that nothing is known about the percentage to be estimated.

Textbook Question

b. Use the information from a previous Harris survey in which 26% of respondents said that they believed in astrology.

Textbook Question

Airline Seating You are the operations manager for American Airlines and you are considering a higher fare level for passengers in aisle seats. You want to estimate the percentage of passengers who now prefer aisle seats. How many randomly selected air passengers must you survey? Assume that you want to be 95% confident that the sample percentage is within 2.5 percentage points of the true population percentage.

a. Assume that nothing is known about the percentage of passengers who prefer aisle seats.

Textbook Question

Online Gambling Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be 99% confident that your estimate is in error by no more than two percentage points?

b. Assume that 18% of all adults gamble online (based on 2017 data from a Gambling Commission study in Great Britain).

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%.

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