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Understanding Sample Proportions and Normal Approximation in Statistics

In this chapter, we delve into sample proportions and the rules of thumb for determining when to apply normal approximation. Using Christopher Columbus's discovery year as a backdrop, we analyze a Gallup Poll where 210 out of 501 respondents accurately identified this fact. Key rules of thumb are explained, such as the conditions required for using population standard deviation and the normal approximation for sample distributions. Examples illustrate how to standardize p-hat values, assess survey undercoverage, and calculate probabilities using statistical tools.

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Understanding Sample Proportions and Normal Approximation in Statistics

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  1. Chapter 9.2: Sample Proportion Mr. Lynch AP Statistics

  2. Sample Proportions • In what year did Christopher Columbus “discover” America? • What is our classes p-hat? • 210 of the 501 sampled in a Gallup Poll knew this.

  3. Rules of Thumb • RULE OF THUMB 1: • You can only use the standard deviation formula of the population is large enough. Specifically N 10n • RULE OF THUMB 2: • You can only use A NORMAL APPROXIMATION to the propoprtion sampling distribution with values of n and p if np10 AND nq10

  4. The Normal Approximation • When n is LARGE, the Sampling Distribution of is approximately normal • EXAMPLE 9.7: APPLYING TO COLLEGE • SRS OF n = 1500; p = .35 • Check Rules of Thumb • P(the random sample will produce a percentage within 2% of the actual population proportion of 35%)

  5. The Normal Approximation • Draw it! • Standardize the p-hat values • Re-draw with the new z-scores • Calculate Area with table or normalcdf

  6. SURVEY UNDERCOVERAGE • EXAMPLE 9.8 SURVEY UNDERCOVERAGE? • 11% US adults are black • A recent national sample of 1500 adults had only a 9.2% black representation • Should we suspect racial discrimination or some sort of under-coverage?

  7. SURVEY UNDERCOVERAGE • Rules of Thumb • Draw it! • Standardize the p-hat value • Re-draw with the new z-score • Calculate Area with table or normalcdf

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