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Section 7.2

Section 7.2. Sampling Distribution of the Sample Mean. For a finite population , the sampling distribution of the sample mean is the. For a finite population, the sampling distribution of the sample mean is the set of means from all possible samples of a specific size , n.

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Section 7.2

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  1. Section 7.2 Sampling Distribution of the Sample Mean

  2. For a finitepopulation, the sampling distribution of the sample mean is the

  3. For a finite population, the sampling distribution of the sample mean is the set of means from all possible samples of a specific size, n.

  4. Simulated Sampling Distribution 4 steps to construct:

  5. Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population

  6. Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population 2) Compute summary statistic for sample

  7. Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population 2) Compute summary statistic for sample 3) Repeat steps 1 and 2 many times

  8. Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population 2) Compute summary statistic for sample 3) Repeat steps 1 and 2 many times 4) Display distribution of summary statistic

  9. Find mean and standard deviationusing Freq. Table of Population (not a sample distribution) Page 428

  10. Find mean and standard deviationusing Freq. Table of Population (not a sample distribution) L2 L1

  11. Find mean and standard deviationusing Freq. Table of Population (not a sample distribution) L1 L2 STAT CALC 1-Var Stats L1, L2

  12. Find mean and standard deviationusing Freq. Table of Population (not a sample distribution)

  13. Find mean and standard deviationusing a sample distribution(not of population) Suppose we make 5 sampling distributions of the sample mean for samples of size: • n = 1 • n = 4 • n = 10 • n = 20 • n = 40

  14. Find mean and standard deviationusing a sample distribution, n=1(from a population) Note: this column is a Sample Distribution using a sample size n=1 (looks just like the Population Frequency Graph) Note: this column is of a Population

  15. Using Population Freq. Table, generate Sampling Distributions of different sample sizes (next slides) n = 4 n = 10 n = 20 n = 40 4 Steps not shown…see Page 428 & 429 in textbook for calculations when n = 4 Note: this graph is of a Population

  16. To clarify Step 1 in textbook, page 428, we can use a random table of digits OR a calculator to assign digits to possible outcomes Using random table of digits: 001- 524 525 - 725 726 - 904 905 - 974 975 – 999, 1000 Or using Calculator: MATH PRB randInt (1, 1000)

  17. Results After repeating Steps 1-4 many times (i.e. Fathom) using different size samples, the results are:

  18. As n increases, what happens to: • Shape • Center • Spread

  19. As n increases, what happens to: • Shape: more normal • Center • Spread

  20. As n increases, what happens to: • Shape: more normal • Center: stays same • Spread

  21. As n increases, what happens to: • Shape: more normal • Center: stays same • Spread: decreases

  22. Common Symbols Note: a calculator cannot tell if a list (L1 or L2) is from a population or sample distribution…you have to know which is which. Page 430

  23. Properties of Sampling Distribution of the Sample Mean These properties depend on using ________ samples.

  24. Properties of Sampling Distribution of the Sample Mean These properties depend on using random samples.

  25. Properties of Sampling Distribution of the Sample Mean These properties depend on using random samples. These properties will cover center, spread, and shape.

  26. Properties of Sampling Distribution of the Sample Mean If a random sample of size n is selected from a population with mean and standard deviation , then:

  27. Properties of Sampling Distribution of the Sample Mean Center The mean, x, of the sampling distribution of x equals the mean of the population, : x =

  28. Properties of Sampling Distribution of the Sample Mean Center The mean, x, of the sampling distribution of x equals the mean of the population, : x = In other words, the means of random samples are centered at the population mean.

  29. Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. x =

  30. Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. X = When sample size increases, spread ________.

  31. Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. X = When sample size increases, spread decreases

  32. Properties of Sampling Distribution of the Sample Mean Shape The shape of the sampling distribution will be approximately normal if the population is approximately normal.

  33. Properties of Sampling Distribution of the Sample Mean Shape The shape of the sampling distribution will be approximately normal if the population is approximately normal. For other populations, the sampling distribution becomesmore normal as n increases (Central Limit Theorem).

  34. Using These Properties • When can you use property that mean of sampling distribution of the mean is equal to the mean of the population, x = ?

  35. Using These Properties • When can you use property that mean of sampling distribution of the mean is equal to the mean of the population, x = ? Anytime you use random sampling. Shape of pop., size of sample, how large pop. is, or whether sample with or without replacement do not matter.

  36. Using These Properties • When can we use property that standard error of sampling distribution of the mean, x, is equal to ?

  37. Using These Properties With a population of any shape and with any sample size as long as you randomly sample with replacement or

  38. Using These Properties With a population of any shape and with any sample size as long as you randomly sample with replacement or you randomly sample without replacement and the sample size is less than 10% of population size.

  39. Using These Properties 3. When can we assume the sampling distribution is approximately normal?

  40. Using These Properties 3. When can we assume the sampling distribution is approximately normal? If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or _______.

  41. Using These Properties 3. When can we assume the sampling distribution is approximately normal? If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or if we are told sample size is very large.

  42. Using These Properties 3. When can we assume the sampling distribution is approximately normal? If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or if we are told sample size is very large (think Central Limit Theorem).

  43. Using These Properties 4. Isn’t the size of the population really important?

  44. Using These Properties 4. Isn’t the size of the population really important? As long as the sample was randomly selected and as long as the population is much larger than the sample, it does not matter how large the population is.

  45. Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?

  46. Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? Could add up heights of all bars ≤ 1.5

  47. Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? Could add up heights of all bars ≤ 1.5 What shape is this?

  48. Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? How do we find the area under the normal curve?

  49. Average of 1.5 children or fewer? normalcdf(lower bound, upper bound, mean, standard deviation)

  50. Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? Recall, for the population μ = 0.9 and σ = 1.1 What are the sampling distribution mean and standard deviation (standard error)?

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