#### Transcript Chapter 1: Introduction to Statistics

COURSE: JUST 3900 TIPS FOR APLIA Chapter 7: Distribution of Sample Means Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick Key Terms: Don’t Forget Notecards Sampling Error (p. 201) Distribution of Sample Means (p. 201) Sampling Distribution (p. 202) Central Limit Theorem (p. 205) Expected Value of M (p. 206) Standard Error of M (p. 207) Law of Large Numbers (p. 207) Formulas Standard Error of M: 𝜎𝑀 = 𝑀−𝜇 𝜎𝑀 z-Score Formula: 𝑧 = 𝜎 𝑛 = 𝜎2 𝑛 = 𝜎2 𝑛 Central Limit Theorem Question 1: A population has a mean of µ = 50 and a standard deviation of σ = 12. a) b) c) d) For samples of size n = 4, what is the mean (expected value) and the standard deviation (standard error) for the distribution of sample means? If the population distribution is not normal, describe the shape of the distribution of sample means based on n = 4. For samples of size n = 36, what is the mean (expected value) and the standard deviation (standard error) for the distribution of sample means? If the population distribution is not normal, describe the shape of the distribution of sample means based on n = 36. Central Limit Theorem Question 1 Answers: a) Expected Value of M: µ = 50 Standard Error of M: 𝜎𝑀 = b) c) = 12 4 = 12 2 =6 The distribution of sample means does not satisfy either criterion to be normal. It would not be a normal distribution. Expected Value of M: µ = 50 Standard Error of M: 𝜎𝑀 = d) 𝜎 𝑛 𝜎 𝑛 = 12 36 = 12 6 =2 Because the sample size is greater than n = 30, the distribution of sample means is a normal distribution. Understanding the Sampling Distribution of M Question 2: As sample size increases, the value of expected value of M also increases. (True or False?) Understanding the Sampling Distribution of M Question 2 Answer: False. The expected value of M does not depend on sample size; it will always be equal to the population mean: µ. Understanding the Sampling Distribution of M Question 3: As sample size increases, the value of the standard error also increases. (True or False?) Understanding the Sampling Distribution of M Question 3 Answer: False. The standard error decreases as sample size increases. In Question 1a, the standard error was 𝜎𝑀 = 𝜎 𝑛 = 12 4 = 12 2 = 6. However in Question 1c, in which the sample size was increased from n = 4 to n = 36, the standard error decreased: 𝜎𝑀 = 𝜎 𝑛 = 12 36 = 12 6 = 2. Using z-Scores with the Distribution of Sample Means Question 4: For a population with a mean of µ = 40 and a standard deviation of σ = 8, find the z-score corresponding to a sample mean of M = 44 for each of the following sample sizes. a) b) n=4 n = 16 Using z-Scores with the Distribution of Sample Means Question 4 Answers; a) The standard error is 𝜎𝑀 = 𝑧= b) 𝑀−𝜇 𝜎𝑀 = 44−40 4 = 4 4 𝑀−𝜇 𝜎𝑀 = 44−40 2 = 4 2 = 8 4 = 8 16 8 2 = = 4, and = 1.00 The standard error is 𝜎𝑀 = 𝑧= 𝜎 𝑛 𝜎 𝑛 = 2.00 8 4 = = 2, and Using z-Scores with the Distribution of Sample Means Question 5: What is the probability of obtaining a sample mean greater than M = 60 for a random sample of n = 16 scores selected from a normal population with a mean of µ = 65 and a standard deviation of σ = 20? Using z-Scores with the Distribution of Sample Means Question 5 Answer: 𝜎 𝑛 M corresponds to 𝑧 = p(M > 60) = p(z > -1.00) = 0.8413 (or 84.13%) Z = -1.00 = 60−65 5 = 20 4 The standard error is 𝜎𝑀 = 𝑀−𝜇 𝜎𝑀 = 20 16 = −5 5 = 5, = −1.00, Using z-Scores with the Distribution of Sample Means Question 6: A positively skewed distribution has µ = 60 and σ = 8. a) b) What is the probability of obtaining a sample mean greater than M = 62 for a sample of n = 4 scores? What is the probability of obtaining a sample mean greater than M = 62 for a sample of n = 64 scores? Using z-Scores with the Distribution of Sample Means Question 6 Answer: a) The distribution does not satisfy either of the criteria for being normal. Therefore, you cannot use the unit normal table, and it is impossible to find the probability. Remember: A distribution of sample means is normal if at least one of the following condition are met: 1) The population from which the samples are selected is normal, 2) The number of scores (n) in each sample is relatively large, around 30 or more. b) With n = 64, the distribution of sample means is nearly normal. 𝜎 8 8 The standard error is 𝜎𝑀 = = = = 1, 𝑛 M corresponds to 𝑧 = 𝑀−𝜇 𝜎𝑀 = 64 62−60 1 = 8 2 1 = 2.00, p(M > 62) = p(z > 2.00) = 0.0228 (or 2.28%) Three Different Distributions Question 7: A population has a mean of µ = 100 and a standard deviation of σ = 15. A sample of n = 25 scores is taken with a mean of M = 101.2 and a standard deviation of s = 11.5. a) b) c) On average, how much difference should there be between the population mean and a single score selected from this population? On average, how much difference should there be between the sample mean and a single score selected from that sample? On average, how much difference should there be between the population mean and the sample mean of any sample consisting of n = 25 scores? Three Different Distributions Question 7: b) σ = 15 s = 11.5 c) 𝜎𝑀 = a) 𝜎 𝑛 = 15 25 = 15 5 =3 Frequently Asked Questions FAQs What effect does sample size have on the standard error? As sample size increases, standard error decreases. This is because large samples are more representative of the population. Thus we can expect less difference, or error. As sample size decreases, standard error increases. This is because smaller samples are less representative of the population. Thus we can expect a greater difference, or error. Frequently Asked Questions FAQs What’s the difference between standard deviation and standard error? Go back and review the slides for Question 7. The standard deviation deals with means and SCORES. If Aplia asks a question about the average difference between the population mean and a SCORE, we are looking for the population standard deviation, not the standard error. The sample standard deviation deals with the sample mean and scores for a particular sample. If you see the words “mean” and “score,” think standard deviation, not standard error. Standard error, however, measures the average distance between the population mean and any sample mean of size n. If you see the words “sample mean” and “population mean,” without reference to scores, think standard error.