Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.1.35a

Chapter 7, Problem 7.1.35a

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.

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 2.5%, which is 0.025 in decimal form. Since nothing is known about the percentage of passengers who prefer aisle seats, assume 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. Using p = 0.5, Z = 1.96, and E = 0.025, the formula becomes: n = (1.96^2 * 0.5 * (1 - 0.5)) / 0.025^2.

Step 4: Simplify the numerator of the formula. Calculate Z^2 (1.96^2), then multiply it by p (0.5) and (1 - p) (0.5). This gives the value for the numerator.

Step 5: Simplify the denominator of the formula. Square the margin of error (E = 0.025), then divide the numerator by this value 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 Determination

Sample size determination is a statistical method used to calculate the number of observations needed to achieve a desired level of confidence and precision in estimating a population parameter. In this scenario, the goal is to estimate the percentage of passengers who prefer aisle seats with a 95% confidence level and a margin of error of 2.5 percentage points.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the confidence interval derived from the sample. A 95% confidence level indicates that if the same sampling procedure were repeated multiple times, approximately 95% of the calculated intervals would contain the true population percentage of aisle seat preference.

Margin of Error

The margin of error quantifies the range within which the true population parameter is expected to fall, based on the sample estimate. In this case, a margin of error of 2.5 percentage points means that the estimated percentage of aisle seat preference could vary by this amount above or below the sample percentage, affecting the reliability of the findings.

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Finding the Minimum Sample Size Needed for a Confidence Interval

Related Practice

Textbook Question

Controversial Song The song “Baby It’s Cold Outside” generated much controversy because of its lyrics and tone. CBS New York conducted a survey by asking viewers to use the Internet to respond to a question asking whether that song was really too offensive to play. Among 1043 Internet users who chose to respond, 986 said that the song was not too offensive, and 57 of the respondents said that the song was too offensive.

a. Construct a 95% confidence interval estimate of the proportion of the population having the belief that the song is not too offensive.

Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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