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)
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).