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

문제 7.1.37c

7장, 문제 7.1.37c

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?

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

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Step 1: Identify the formula for determining the required sample size for estimating a population proportion. The formula is:

n = (z² × p × (1 - p)) / E²

, where

n

is the sample size,

z

is the z-score corresponding to the confidence level,

p

is the estimated proportion, and

E

is the margin of error.

Step 2: Determine the values for the variables in the formula. For a 90% confidence level, the z-score is approximately 1.645. The margin of error (

E

) is given as 0.03 (three percentage points). Since the estimated proportion (

p

) is not provided, use 0.5 as a conservative estimate to maximize 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 = (1.645² × 0.5 × (1 - 0.5)) / 0.03²

Step 4: Simplify the formula step by step. First, calculate

1.645²

, then multiply it by

0.5 × (1 - 0.5)

, and finally divide the result by

0.03²

. This will give the required sample size.

Step 5: Address the second part of the question. Surveying only the people you know would introduce bias into the sample, as it would not be representative of the worldwide population. A random sampling method should be used to ensure the results are generalizable to the entire population.

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Sample Size Determination

Sample size determination is a statistical process 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 ensure that the estimated percentage of those intending to purchase the new iPhone is accurate within three percentage points, with 90% confidence.

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 repeated 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

Margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It indicates the range within which the true population parameter is expected to lie, based on the sample data. In this scenario, a margin of error of three percentage points means that the actual percentage of consumers intending to purchase the new iPhone could be three points higher or lower than the survey estimate.

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