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AP Statistics

AP Statistics. Chapter 9 Notes. Ch 9 Vocabulary. parameter: a number that describes the population statistic: a number that can be computed from the sample data without making use of any unknown parameters. µ  population mean  sample mean σ  population standard deviation

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AP Statistics

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  1. AP Statistics Chapter 9 Notes

  2. Ch 9 Vocabulary • parameter: a number that describes the population • statistic: a number that can be computed from the sample data without making use of any unknown parameters. • µ  population mean •  sample mean • σ  population standard deviation • s  sample standard deviation • p population proportion •  sample proportion

  3. Ch 9 Vocab continued • Sampling variability: The value of a statistic varies in repeated random sampling. • Sampling Distribution: The distribution of the values taken by a statistic in all possible samples of the same size from the same population. • (Very important to understand)

  4. Bias and Variability • A statistic is an unbiased estimator of a parameter if the mean of its sampling distribution is equal to the true value of the parameter being estimated. • The variability of a statistic is described by the spread of its sampling distribution. • Bigger sample size  smaller spread • Population size does not matter

  5. Sampling Distribution of a Sample mean (x-bar) Mean (μ ) = μ Standard deviation (σ) = • Only use if N > 10n • If an SRS of size n is taken from a population that is Normally distributed, then the sampling distribution is also Normal.

  6. Central Limit Theorem • Draw an SRS of size n from any population with mean μ and standard deviation σ. When n is large, the sampling distribution of the sample mean, is close to the Normal distribution….

  7. Example • Assume IQ scores are Normally distributed with a mean of 100 and a standard deviation of 15. • 1. What is the probability of a randomly selected person having an IQ score of more than 120? • 2. What is the probability that a random sample of 7 people will have a mean IQ score of more than 120?

  8. Example 2 • Assume test scores for a large population have a mean of 72 and standard deviation of 8. • 1. What is the probability a randomly selected person has a test score of less than 70? • 2. Take a random sample of 40 people. What is the probability their mean score is less than 70?

  9. Trends to remember • Means of random samples are less variable than individual observations. • Means of random samples are more Normal than individual observations.

  10. Sampling Distribution of a Sample Proportion ( ) • Shape: approximately Normal (see Rule on following slide). • Mean (μ ) = p • Std Dev (σ ) = • n  size of SRS • p  population proportion

  11. Rules for Applying formulas • Rule #1: aka Independence Rule • The formula for standard deviation only applies if the individuals in the sample are independent. This occurs if the population is at least 10 times bigger than the sample. (N > 10n)

  12. Rules for applying formulas • Rule #2: aka Normality Rule • For proportions, the sampling distribution is approximately Normal if np > 10 and n(1-p) > 10 • For means, the sampling distribution is… • Normal is the population is Normally distributed. • approximately Normal if the sample size n is large enough. (We usually say n needs to be > 30).

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