Agenda

1. Agenda • Spørgsmål til projekt 2 • Sampling distribution

2. Learning Objectives • Statistic vs. Parameter • Sampling Distributions • Mean and Standard Deviation of the Sampling Distribution of a Proportion • Standard Error • Sampling Distribution Example

3. Population Sample Learning Objective 1:Statistic and Parameter • A statistic is a numerical summary of sample data such as a sample proportion or sample mean. • A parameter is a numerical summary of a population such as a population proportion or population mean. • In practice, we seldom know the values of parameters. • We use sample data to estimate parameters. • Our estimates are called statistics. • Hvad er følgende? • μ • s • σ

4. Learning Objective 2:Sampling Distributions Example: • Prior to counting the votes, the proportion in favor of Mr. Barack Obama is an unknown parameter. • An exit poll of 3.160 voters reported that the sample proportion in favor of a recall was 0.54. • If a different random sample of about 3.000 voters were selected, a different sample proportion would occur. Hvad betyder følgende (definition)? The sampling distribution of the sample proportion shows all possible values and the probabilities for those values.

5. Learning Objective 2:Sampling Distributions • The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take. • Sampling distributions describe the variability that occurs from study to study using statistics to estimate population parameters

6. Learning Objective 3:Mean and SD of the Sampling Distribution of a Proportion • For a random sample of size n from a population with proportion p of outcomes in a particular category, the sampling distribution of the proportion of the sample in that category has

7. Learning Objective 4:The Standard Error • To distinguish the standard deviation of a sampling distribution from the standard deviation of an ordinary probability distribution, we refer to it as a standard error. • Example: If the population proportion supporting the reelection of Mr. Obama was 0.50, would it have been unlikely to observe the exit-poll sample proportion of 0.565?

8. Learning Objective 5:Example: 2012 Election • An exit poll had 2.705 people • Assume 50% support the reelection of Mr. Obama • Find the estimate of the population proportion and the standard error.

9. Learning Objective 5:Example: 2012 Election

10. Learning Objective 5:Example: 2006 California Election • The sample proportion of 0.565 is more than six standard errors from the expected value of 0.50. • The sample proportion of 0.565 voting for reelection of Mr. Obama would be very unlikely if the population proportion were p = 0.50

11. Learning Objectives • The Sampling Distribution of the Sample Mean • Effect of n on the Standard Error • Central Limit Theorem (CLT)

12. Learning Objective 1:The Sampling Distribution of the Sample Mean • The sample mean, x, is a random variable. • The sample mean varies from sample to sample. • By contrast, the population mean, µ, is a single fixed number.

13. Learning Objective 1:The Sampling Distribution of the Sample Mean • For a random sample of size n from a population having mean µ and standard deviation σ, the sampling distribution of the sample mean has: • Center described by the mean µ (the same as the mean of the population). • Spread described by the standard error, which equals the population standard deviation divided by the square root of the sample size: • standard error of

14. Learning Objective 2:Effect of n on the Standard Error • Knowing how to find a standard error gives us a mechanism for understanding how much variability to expect in sample statistics “just by chance”. • The standard error of the sample mean = • As the sample size n increases, the standard error decreases. • With larger samples, the sample mean is more likely to fall closer to the population mean.

15. Learning Objective 3:Central Limit Theorem (CLT) • For random sampling with a large sample size n, the sampling distribution of the sample mean is approximately a normal distribution. • This result applies no matter what the shape of the probability distribution from which the samples are taken.

16. Learning Objective 3:CLT: Impact of increasing n

17. Learning Objective 3:CLT: How Large a Sample? • The sampling distribution of the sample mean takes more of a bell shape as the random sample size n increases. • The more skewed the population distribution, the larger n must be before the shape of the sampling distribution is close to normal. • In practice, the sampling distribution is usually close to normal when the sample size n is at least about 30. • If the population distribution is approximately normal, then the sampling distribution is approximately normal for all sample sizes.