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From Last week. Q22. A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers What is the probability of guessing correctly for any question?. Q22.

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From Last week

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  1. From Last week

  2. Q22 • A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers • What is the probability of guessing correctly for any question?

  3. Q22 • A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers • On average, how many questions would a student get correct for the entire test?

  4. Q22 • A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers • What is the probability that a student would get more than 15 answers correct simply by guessing?

  5. Q22 • A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers • What is the probability that a student would get 15 or more answers correct simply by guessing?

  6. The Distribution of Sample Means

  7. Review • z-scores close to zero indicate that the sample mean is relatively close to the population mean • z-scores beyond 2 represent extreme values that are quite different from the population mean

  8. The Distribution of Sample Means • Definition: the set of means from all the possible random samples of a specific size (n) selected from a specific population • Theoretical distribution because normally you wouldn’t have all the information about a population available, and that’s why we have to use samples to make inferences about the population

  9. Characteristics of the distribution of sample mean • The sample means tend to cluster around the population mean • The distribution of sample means can be used to answer probability questions about sample means • The distribution of sample means is approximately normal in shape

  10. Central Limit Theorem 1. The mean of the distribution of sample means is called the Expected Value of M 2. The standard deviation of the distribution of sample means is called the Standard Error of M 3. The shape of the distribution of sample means tends to be normal.

  11. Example • Imagine a population that is normally distributed with µ=110 and σ=24 • If we take a sample from this population, how accurately would the sample mean represent the population mean?

  12. q6 • For a population with a mean of µ=70 and a σ=20, how much error, on average, would you expect between the sample mean and the population mean for each of the following sample sizes • N=4 scores

  13. q6 • For a population with a mean of µ=70 and a σ=20, how much error, on average, would you expect between the sample mean and the population mean for each of the following sample sizes • N=16 scores

  14. q6 • For a population with a mean of µ=70 and a σ=20, how much error, on average, would you expect between the sample mean and the population mean for each of the following sample sizes • N=25 scores

  15. q8 • If the population standard deviation is 8, how large a sample is necessary to have a standard error that is? • Less than 4 points?

  16. q8 • If the population standard deviation is 8, how large a sample is necessary to have a standard error that is? • Less than 2 points?

  17. q8 • If the population standard deviation is 8, how large a sample is necessary to have a standard error that is? • Less than 1 point?

  18. Probability and Sample Means • Because the distribution of sample means tends to be normal, the z-score value obtained for a sample mean can be used with the unit normal table to obtain probabilities.

  19. z-Scores and Location within the Distribution of Sample Means • Within the distribution of sample means, the location of each sample mean can be specified by a z-score, M – μ z = ───── σM

  20. Example • GRE quantitative scores are considered to be normally distributed with a  = 500 and  = 100. • An exceptional group of 16 graduate school applicants had a mean GRE quantitative score of 710. • What is the probability of randomly selecting 16 graduate school applicants with an even greater mean GRE quantitative score? • p (> 710 ) = ?

  21. q11 • A sample of n=4 scores has a mean of m=75. Find the z-score for this sample • If it was obtained from a population with  = 80 and  = 10

  22. q11 • A sample of n=4 scores has a mean of m=75. Find the z-score for this sample • If it was obtained from a population with  = 80 and  =40

  23. q14 • The population IQ scores forms a normal distribution, with a mean of 100 and standard deviation of 15. What is the probability of obtaining a sample mean greater than M=97 • For a random sample of n=9 people?

  24. q14 • The population IQ scores forms a normal distribution, with a mean of 100 and standard deviation of 15. What is the probability of obtaining a sample mean greater than M=97 • For a random sample of n=25 people?

  25. Example 2 • Scores on a test form a normal distribution with µ=70 and σ=12. With a sample size of n=16. • What is the probability of obtaining a sample of at least 75? 70 0 75

  26. What proportion of the sample means will be lower than 73?

  27. What is the probability of obtaining a sample with a mean less than 64?

  28. What proportion of the sample means will be within 2 points of the population mean?

  29. Example 3 • A normal distribution with µ=80 and σ=15, sample size of n=36. • What sample means would mark off the most extreme 5% of the distribution? • What should you do?

  30. q17 • A population of scores forms a normal distribution with a mean of µ=80 and σ=10 • What proportion of the scores have values between 75 and 85?

  31. q17 A population of scores forms a normal distribution with a mean of µ=80 and σ=10 For samples of n=4, what proportion of the samples will have means between 75 and 85?

  32. q17 A population of scores forms a normal distribution with a mean of µ=80 and σ=10 For samples of n=16, what proportion of the samples will have means between 75 and 85?

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