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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 6 - Confidence IntervalsProblema 6.3.19a
Capitolo 6, Problema 6.3.19a

Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed.

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Step 1: Recall the formula for determining the minimum sample size for estimating a population proportion when no preliminary estimate is available: n = (Z^2 * 0.25) / E^2. Here, Z is the critical value corresponding to the desired confidence level, and E is the margin of error.
Step 2: Identify the given values. The confidence level is 90%, so the critical value Z can be found using a Z-table or standard normal distribution. For a 90% confidence level, Z ≈ 1.645. The margin of error E is given as 0.03 (3%).
Step 3: Substitute the values into the formula. Use Z = 1.645, E = 0.03, and the maximum variability for the population proportion (p = 0.5, q = 1 - p = 0.5, so p * q = 0.25). The formula becomes n = (1.645^2 * 0.25) / 0.03^2.
Step 4: Simplify the numerator by squaring the Z-value and multiplying it by 0.25. Then, simplify the denominator by squaring the margin of error (E).
Step 5: Divide the simplified numerator by the simplified denominator to calculate the minimum sample size. Always round up to the nearest whole number, as sample size must be an integer.

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Population Proportion

The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, U.S. families who eat fast food at least once per week. It is denoted by 'p' and is crucial for estimating how widespread a behavior or opinion is within a defined group.
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Constructing Confidence Intervals for Proportions

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to achieve a desired level of precision in estimates. In this scenario, it involves using the desired confidence level (90%) and margin of error (3%) to ensure that the sample accurately reflects the population proportion.
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Sampling Distribution of Sample Proportion

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For this question, a 90% confidence interval means that if the same sampling method were repeated multiple times, 90% of the intervals would contain the true population proportion.
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Introduction to Confidence Intervals
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