1. Exams 2. Sampling Distributions 3. Estimation + Confidence Intervals

1. 1. Exams2. Sampling Distributions3. Estimation + Confidence Intervals

2. INFERENTIAL STATISTICS • Samples are only estimates of the population • Sample statistics will be slightly off from the true values of its population’s parameters • Sampling error: • The difference between a sample statistic and a population parameter • Probability theory • Permits us to estimate the accuracy or representativeness of the sample

3. The “Catch-22” of Inferential Statistics • When we collect a sample, we know nothing about the population’s distribution of scores • We can calculate the mean (x-bar) & standard deviation (s) of our sample, but  and  are unknown • The shape of the population distribution (normal?) is also unknown

4. Probability Theory Allows Us To Answer: What is the likelihood that a given sample statistic accurately represents a population parameter? SampleN = 150 μ = ??? (N= Thousands) X=9.6 Number of serious crimes committed in year priorto prison for inmates entering the prison system

5. Sampling Distribution(a.k.a. “Distribution of Sample Outcomes”) • “OUTCOMES” = proportions, means, etc. • From repeated random sampling, a mathematical description of all possible sampling event outcomes • And the probability of each one • Permits us to make the link between sample and population… • Answer the question: “What is the probability that a sample finding is due to chance?”

6. Relationship between Sample, Sampling Distribution & Population • Empirical (exists in reality) • but unknown • Nonempirical (theoretical or hypothetical) • Laws of probability allow us • to describe its characteristics • (shape, central tendency, • dispersion) • Empirical & known (e.g., • distribution shape, mean, standard deviation)

7. Sampling Distribution: Characteristics • Central tendency • Sample means will cluster around the population mean • Since samples are random, the sample means should be distributed equally on either side of the population mean • The mean of the sampling distribution is always equal to the population mean • Shape: Normal distribution • Central Limit Theorem: • Regardless of the shape of a raw score distribution (sample or population) of an interval-ratio variable, the sampling distribution will be approximately normal, as long as sample size is ≥ 100

8. Sampling Distribution: Characteristics • Dispersion: Standard Error (SE) • Measures the spread of sampling error that occurs when a population is sampled repeatedly • Same thing as standard deviation of the sampling distribution • Tells exactly how much error, on average, should exist between the sample mean & the population mean • Formula: σ / √N • However, because σ usually isn’t known, s (sample standard deviation) is used to estimate population standard deviation

9. Sampling Distribution • Standard Error • Law of Large Numbers: The larger the sample size (N), the more probable it is that the sample mean will be close to the population mean • In other words: a big sample works better (should give a more accurate estimate of the pop.) than a small one • Makes sense if you study the formula for standard error

10. 1. Estimation ESTIMATION

11. Introduction to Estimation • Estimation procedures • Purpose: • To estimate population parameters from sample statistics • Using the sampling distribution to infer from a sample to the population • Most commonly used for polling data • 2 components: • Point estimate • Confidence intervals

12. Point Estimate: Value of a sample statistic used to estimate a population parameter Confidence Interval: A range of values around the point estimate Estimation Point Estimate Confidence Interval .58 .546 .614 Confidence Limit (Lower) Confidence Limit (Upper)

13. Example • CNN Poll (CNN.com; Feb 20, 2009): Slight majority thinks stimulus package will improve economy • “The White House's economic stimulus plan isn't a surefire winner with the American public, but a majority does think the recovery plan will help. According to a new poll, fifty-three percent said the plan will improve economic conditions, while 44 percent said it won't stimulate the economy.” • “On an individual level, there was less hope for improvement. According to the poll, 67 percent said it would not help them personally.” • “The Poll was conducted Wednesday and Thursday (Feb 18-19, 2009), with 1,046 people questioned by telephone. The survey's sampling error is plus or minus 3 percentage points.”

14. Estimation • POINT ESTIMATES • (another way of saying sample statistics) • CONFIDENCE INTERVAL • a.k.a. “MARGIN OF ERROR” • Indicates that over the long run, 95 percent of the time, the true pop. value will fall within a range of +/- 3 • Point estimates & confidence interval should be reported together “…but a majority does think the recovery plan will help, according to a new poll. Fifty-three percent said the plan will improve economic conditions, while 44 percent said it won't stimulate the economy. …. The Poll was conducted Wednesday and Thursday (Feb 18-19, 2009), with 1,046 people questioned by telephone. The survey's sampling error is plus or minus 3 percentage points.

15. Estimation1 : Pick Confidence Level • Confidence LEVEL • Probability that the unknown population parameter falls within the interval • Alpha () • The probability that the parameter is NOT within the interval • Confidence level = 1 -  • Conventionally, confidence level values are almost always 95%or 99%

16. 2. Divide the probability of error equally into the upper and lower tails of the distribution (2.5% error in each tail with 95% confidence level) Find the corresponding Z score Procedure for Constructing an Interval Estimate 0.95 .025 .025 -1.96 1.96  Z scores 

17. Procedure for Constructing an Interval Estimate 3. Construct the confidence interval • Proportions (like the eavesdropping poll example): • Sample point estimate (convert % to a proportion): • “Fifty-three percent said the plan will improve economic conditions…” • 0.53 • Sample size (N) = 1,046 • Formula 7.3 in Healey • Numerator = (your proportion) (1- proportion) • 95% confidence level (replicating results from article) • 99% confidence level – intervals widen as level of confidence increases

18. Example 1: Estimate for the economic recovery poll • p = .53 (53% think it will help) • Z = 1.96 (95% confidence interval) • N = 1046 (sample size) • What happens when we… • Recalculate for N = 10,000 • N back to original, recalculate for p. = .90 • Back to original, but change confidence level to 99%

19. Example 2 • Houston Chronicle (2008) — A University of Texas poll to be released today shows Republican presidential candidate John McCain and GOP Sen. John Cornyn leading by comfortable margins in Texas, as expected. But the statewide survey of 550 registered voters has one very surprising finding: 23 percent of Texans are convinced that Democratic presidential nominee Barack Obama is a Muslim. • The Obama-is-a-Muslim confusion is caused by fallacious Internet rumors and radio talk-show gossip. McCain went so far at one of his town hall meetings to grab a microphone from a woman who claimed that Obama was an Arab. • GIVEN THIS INFO, IDENTIFY A POINT ESTIMATE & CALCULATE THE CONFIDENCE INTERVAL (ASSUMING A 95% CONFIDENCE LEVEL). • CALCULATE THE CONFIDENCE INTERVAL ASSUMING A 99% CONFIDENCE LEVEL

20. A Good Estimate is Unbiased • Sample means and proportions (like the .53 [53%] & .23 [23%]) are UNBIASED estimates of the population parameters • We know that the mean of the sampling distribution = the pop. Mean • Other sample statistics (such as standard deviation) are biased • The standard deviation of a sample is by definition smaller than the standard deviation of the population • Bottom line: A good estimate is UNBIASED • Trustworthy estimator of the pop. parameter

21. A Good Estimate is Efficient • Efficiency • Refers to the extent to which the sampling distribution is clustered about its mean • Efficiency depends largely on sample size As the sample size increases, the sampling distribution gets tighter (more narrow) • BOTTOM LINE: THE LESS SPREAD (THE SMALLER THE S.E.), THE BETTER

22. Estimation of Population Means • EXAMPLE: A researcher has gathered information from a random sample of 178 households. Construct a confidence interval to estimate the population mean at the 95% level: • An average of 2.3 people reside in each household. Standard deviation is .35.

23. PROCEDURE FOR CONSTRUCTING ANINTERVAL ESTIMATE • A random sample of 429 college students was interviewed • They reported they had spent an average of $178 on textbooks during the previous semester. If the standard deviation (s) of these data is $15 construct an estimate of the population at the 95% confidence level. • They reported they had missed 2.8 days of class per semester because of illness. If the sample standard deviation is 1.0, construct an estimate of the population mean at the 99% confidence level. • Two individuals are running for mayor of Duluth. You conduct an election survey of 100 adult Duluth residents 1 week before the election and find that 45% of the sample support candidate Long Duck Dong, while 40% plan to vote for candidate Singalingdon. • Using a 95% confidence level, based on your findings, can you predict a winner?

24. What influences confidence intervals? • The width of a confidence interval depends on three things • : The confidence level can be raised (e.g., to 99%) or lowered (e.g., to 90%) • N: we have more confidence in larger sample sizes so as N increases, the interval decreases • Variation: more variation = more error • % agree closer to 50% • Higher standard deviations