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Sampling Distributions & Confidence Intervals: Central Limit Theorem Overview

Learn about sampling distributions, Central Limit Theorem, confidence intervals, and probabilities with practical examples and calculations in STAT 211 at West Virginia University. Understand how to estimate population means and construct confidence intervals.

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Sampling Distributions & Confidence Intervals: Central Limit Theorem Overview

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  1. Lecture 8 Dan Piett STAT 211-019 West Virginia University

  2. Last Week • Continuous Distributions • Normal Distributions • Normal Probabilities • Normal Percentiles

  3. Overview • 8.1 Distribution of X-bar and the Central Limit Theorem • 8.2 Large Sample Confidence Intervals for Mu • 8.3 Small Sample Confidence Intervals for Mu

  4. Section 8.1 The Sampling Distribution of the Sample Mean and the Central Limit Theorem

  5. Distribution of the Mean • Suppose we generate multiple samples of size n from a population, we will get a sample mean from each group. • These sample means will have their own distribution. • The sample mean is a random variable with it’s own mean and standard deviation aka. Standard error. (See notation on board) • This is known as the Sampling Distribution of the Sample Mean • Some questions to think about: • What is the shape of the sampling distribution? • What is the mean and standard error of the sampling distribution?

  6. The Central Limit Theorem • The distribution of the sample mean is determined by the shape of the distribution of X • X is Normal • The distribution of the sample mean is normal • Mean mu • (The same as the mean of X) • Standard error sigma/sqrt(n) • (The standard deviation of X divided by the sample size) • What if X is not normal?

  7. Central Limit Theorem Contd • So what if X is not Normal. Assume X~? • The shape of X will depend on the sample size • If n<20 • We cannot be certain the distribution of the sample mean. It is not necessarily normal or even approximately normal • If n≥20 • The Central Limit Theorem States that the distribution of the sample mean will approach normality • Mean mu • (The same as the mean of X) • Standard error sigma/sqrt(n) • (The standard deviation of X divided by the sample size)

  8. Examples • Give the sampling distribution of the sample mean for the following distributions: • X is normally distributed with a mean of 50 and a standard deviation of 20. What is the distribution of the sample mean of n = 25 • X is Exponentially distributed with mean of 20 and standard deviation of 10. What is the distribution of the sample mean of n = 100? • X is normally distributed with a mean of 100 and a standard deviation of 18. What is the distribution of the sample mean of n = 9

  9. Probabilities • Since the distribution of the sample mean follows a normal distribution (under the appropriate conditions) we can calculate probabilities much like last week. • All methods are exactly the same except now we calculate the z-score using our new mean and standard error.

  10. Example: Back to SAT Scores • From last week we said that SAT Math Scores are Normally distributed with a mean of 500 and a standard deviation of 100. • X~N(500,100) • What is the sampling distribution of the sample mean of a class of 25 students? • What is the probability that A RANDOMLY SELECTED STUDENT scores above a 600 on the SAT Math section? • What is the probability that THE MEAN SCORE OF THE 25 STUDENTS is above 600?

  11. Section 8.2 Large Sample Confidence Intervals

  12. Point Estimator • Suppose we do not know the true population mean. How can we estimate it? • We could find a sample and use a statistic such as the sample mean to estimate it • This is known as a point-estimate • But it is unlikely that our sample mean is going to exactly match the population mean even under perfect conditions • For this reason, it is better to state that we believe the true mean is between two numbers a and b • a < µ < b • We can predict a and b using a confidence interval

  13. Confidence Intervals • Our confidence intervals will always be of the form: • Sample Statistic ± critical value * error term • For the population mean, our sample statistic is x-bar • Our critical value will be either Z or t (I will explain t later) • Our error term will be the standard error of the sample mean

  14. Large Sample Confidence Intervals • Recall if n is “large” (≥20), X-bar’s dist. is approximately normal with mean mu and standard error sigma/sqrt(n) • 95% CI

  15. Example • Find the 95% confidence interval for the mean SAT Math Score for x-bar = 502, s = 8, n = 36 • 502 ± 1.96*8/6 • (499.387,504.613) • Conclusion: • We are 95% confident that the true mean SAT Math Score is between 499.387 and 504.631 • Always be sure to state your conclusion

  16. Confidence Levels • In the previous slide, we used a confidence level of 95%. • This corresponded to a critical value of 1.96 • We commonly use 3 different confidence levels: • 90% • Critical Value of 1.645 • 95% • Critical Value of 1.96 • 99% • Critical Value of 2.578

  17. Notes on the Error Term • The error term can be effected by 3 different things • The sample size, n • The larger n, the smaller the error term • The standard deviation, s • The smaller s, the smaller the error term • The confidence level • The higher the confidence level, the larger the critical value, the larger the error term • We cannot choose s in practice, but we can choose the confidence level and often n

  18. Section 8.3 Small Sample Confidence Intervals

  19. Small Sample Confidence Intervals • For “large” sample sizes, we have the convenience of knowing that the distribution approaches normality • We can use the Z table (1.645, 1.98, 2.645) • For “small” sample sizes (<20) we have to do a little more work and we must know that X is normal • Our Central Limit Theorem Rules do not apply • Two Cases • The population standard deviation is known • The standard deviation is unknown

  20. Case 1: Sigma is known • Good news! • In this case we handle our confidence intervals in the exact same way we would for a large sample • Example: • Compute a 99% confidence interval for the population mean when x-bar= 43.2, sigma = 18, n = 16 • (31.6, 54.8) • We are 99% confident that the population mean is between 31.6 and 54.8

  21. Case 2: Sigma is unknown • When the population standard deviation is unknown we need to make a slight adjustment to our formula. • The adjustment is in the critical value. Rather than using our Z values (1.645, 1.98, 2.645) we will be using t values • t values come from the t-distribution. You will only need to know the t-distribution for inference, not probability.

  22. T-distribution • t values can be found on Table F • Notes about t • t is mound shaped and symmetric about the mean, 0 • Just like the standard normal • It looks exactly like the standard normal, except with larger tails. • The values of T require a parameter, degrees of freedom, to find the value on the table • Degrees of freedom are equal to n – 1 • As n increases, t approaches Z

  23. Example • Construct a 95% confidence interval for the mean weight of apples (in grams): • x-bar = 183, s = 14.1, n = 16 • First find t • Alpha/2 = .025 ( because of 95% confidence) • df = n-1 = 15 • t = 2.13 • 183±2.13*14.1/sqrt(16) • (175.5 and 190.5) • We are 95% confident that the mean weight of apples is between 175.5 and 190.5 grams

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