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Section 7.2

Section 7.2. Sampling Distribution of the Sum of a Sample. Properties of Sampling Distribution of the Sample Mean.

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Section 7.2

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  1. Section 7.2 Sampling Distribution of the Sum of a Sample

  2. 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 what are the three basic properties of the sampling distribution of the sample mean for center, spread, and shape?

  3. Properties of Sampling Distribution of the Sample Mean Center x =

  4. Properties of Sampling Distribution of the Sample Mean Spread X = Spread decreases as sample size increases

  5. 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 becomes more normal as n increases (Central Limit Theorem).

  6. Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families? µ = 0.9 σ = 1.1

  7. Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families? Reasonably likely values are those in the middle 95% of the sampling distribution.

  8. Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families?

  9. Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families? µ = 0.9, σ = 1.1 For approx. normal distributions, reasonably likely outcomes are those within 1.96 standard errors of the mean.

  10. Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families? 0.9 1.96(0.25) which gives an average number of children between 0.41 and 1.39

  11. Page 438, P8

  12. Page 438, P8 The shape of the sampling distribution will be approximately normal because the population’s distribution is approximately normal.

  13. Page 438, P8 The shape of the sampling distribution will be approximately normal because the population’s distribution is approximately normal. The mean of the sampling distribution will be equal to the population mean of .266.

  14. Page 438, P8 The shape of the sampling distribution will be approximately normal because the population’s distribution is approximately normal. The mean of the sampling distribution will be equal to the population mean of .266. The standard error is:

  15. Think Problems we’ve worked so far involved: • Shape, center, and spread of sampling distributions of means of various sample sizes -- Use the three properties we discussed

  16. Think Problems we’ve worked so far involved: • Probability a random sample of n families will have an average of x children or less normalcdf(lower bound, upper bound, mean of sampling distribution, standard error)

  17. Think Problems we’ve worked so far involved: • What average numbers of ??? are reasonably likely in a random sample of n things? μx ± 1.96(SE)

  18. Think What if we have problems involving sample totalssuch as if you pick 15 households at random, what is the probability that they have at least 30 motor vehicles among them?

  19. Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size nis selected from a distribution with mean and standard deviation , then:

  20. Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size nis selected from a distribution with mean and standard deviation , then: (1) The mean of the sampling distribution of the sum is sum = n

  21. Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size nis selected from a distribution with mean and standard deviation , then: (2) The standard error of the sampling distribution of the sum is sum =

  22. Properties of the Sampling Distribution of the Sum of a Sample Three properties based on the following premise: If a random sample of size nis selected from a distribution with mean and standard deviation , then: (3) The shape of the sampling distribution will be approximately normal if the population is approximately normally distributed. For other populations the sampling distribution will become more normal as n increases.

  23. The distribution of the number of motor vehicles per household in the U.S. is roughly symmetric with mean 1.7 and standard deviation 1.0.

  24. The distribution of the number of motor vehicles per household in the U.S. is roughly symmetric with mean 1.7 and standard deviation 1.0. (a) If you pick 15 households at random, what is the probability that they have at least 30 motor vehicles among them?

  25. The distribution of the number of motor vehicles per household in the U.S. is roughly symmetric with mean 1.7 and standard deviation 1.0. • If you pick 15 households at random, what is the probability that they have at least 30 motor vehicles among them? Use normalcdf (lower bound, upper bound, mean of the sum, standard deviation of sum)

  26. The distribution of the number of motor vehicles per household in the U.S. is roughly symmetric with mean 1.7 and standard deviation 1.0. • If you pick 15 households at random, what is the probability that they have at least 30 motor vehicles among them? normalcdf (30, 1E99, nμ, ●σ )

  27. The distribution of the number of motor vehicles per household in the U.S. is roughly symmetric with mean 1.7 and standard deviation 1.0. • If you pick 15 households at random, what is the probability that they have at least 30 motor vehicles among them? normalcdf (30, 1E99, (15 ● 1.7), ( ) ) ≈ 0.1226

  28. The distribution of the number of motor vehicles per household in the U.S. is roughly symmetric with mean 1.7 and standard deviation 1.0. (b) If you pick 20 households at random, what is the probability that they have between 25 and 30 motor vehicles among them?

  29. The distribution of the number of motor vehicles per household in the U.S. is roughly symmetric with mean 1.7 and standard deviation 1.0. (b) If you pick 20 households at random, what is the probability that they have between 25 and 30 motor vehicles among them? normalcdf (25, 30, (20 ● 1.7),( ) ) ≈ 0.1635

  30. Suppose a population distribution of the number of children per family in the U.S. has a mean of 0.9 and a standard deviation of 1.1. (a) Do you think it is reasonably likely that a sample of 1000 households will produce at least 1000 children?

  31. Suppose a population distribution of the number of children per family in the U.S. has a mean of 0.9 and a standard deviation of 1.1. • Do you think it is reasonably likely that a sample of 1000 households will produce at least 1000 children? normalcdf (1000, 1E99, (1000●0.9), ) ≈ 0.002; so not reasonably likely

  32. Suppose a population distribution of the number of children per family in the U.S. has a mean of 0.9 and a standard deviation of 1.1. • Do you think it is reasonably likely that a sample of 1000 households will produce at least 1000 children? normalcdf (1000, 1E99, (1000●0.9), ) ≈ 0.002; so not reasonably likely Need P(> 1000 children) to be > 0.025

  33. Suppose a population distribution of the number of children per family in the U.S. has a mean of 0.9 and a standard deviation of 1.1. (b) Suppose now we have a random sample of 1200 households. Does this dramatically improve the chances of seeing at least 1000 children in the sampled households?

  34. Suppose a population distribution of the number of children per family in the U.S. has a mean of 0.9 and a standard deviation of 1.1. (b) Suppose now we have a random sample of 1200 households. Does this dramatically improve the chances of seeing at least 1000 children in the sampled households? Yes normalcdf(1000, 1E99, (1200 ● 0.9), ) ≈ 0.982

  35. Page 438, P10

  36. Page 438, P10

  37. Page 438, P10

  38. Page 438, P10

  39. Page 438, P10

  40. Page 440, E15

  41. Page 440, E15

  42. Page 440, E15 The population standard deviation is 2.402. Now do part b.

  43. Page 440, E15

  44. Page 440, E15

  45. Page 440, E15 d. The rule that about 95% of the observations lie within approximately two standard errors of the population mean works well for n = 25 as this distribution is approximately normal.

  46. Page 443, E21

  47. Page 443, E21(a)

  48. Page 443, E21(b) normalcdf(510, 1E99, 500, 50) ≈ .4207

  49. Page 443, E21(c) normalcdf(510, 1E99, 500, 20) ≈ .3085

  50. Questions?

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