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Chapter 5

Chapter 5. Sampling Distributions. Sampling Distributions. Data are summarized by statistics (mean, standard deviation, median, quartiles, correlation, etc..) Statistics are random variables with a distribution (called sampling distribution). Notation.

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Chapter 5

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  1. Chapter 5 Sampling Distributions

  2. Sampling Distributions • Data are summarized by statistics (mean, standard deviation, median, quartiles, correlation, etc..) • Statistics are random variables with a distribution (called sampling distribution)

  3. Notation • Sample proportion: = (# in group)/(total) • Population proportion: p • Example: At UNCW, p=0.45 • A student obtains a random sample of size 50, of which 30 are female. What is the sample proportion? • = 30/50 = 3/5 or 0.6

  4. Sampling Distribution of • IF data are obtained from a SRS and np>10 and n(1-p)>10, then the sampling distribution of has the following form: • is approximately normal with mean p and standard deviation • Why are we interested in this? • Because we can standardize values of and use tables to find probabilities.

  5. Standardizing sample proportions • Z = (observation – mean)/(standard deviation) • Example: According to a 2002 University of Michigan survey, only about one-third of Americans expected the next 5 years to bring continuous good times (New York Times, Nov 11, 2002). Assume that 33% of the current population of all Americans hold this opinion. Let be the proportion in a random sample of 800 Americans who will hold this opinion. Find the probability that the value of is between 0.35 and 0.37.

  6. Sampling distribution of the sample proportion • Z=(0.35-0.33)/0.01662 = 1.2 • Z=(0.37-0.33)/0.01662=2.41 • Now we have P(1.2<Z<2.41)=0.992-0.8849 = .1071

  7. Another example • Maureen Webster, who is running for mayor in a large city, claims that she is favored by 53% of all eligible voters of that city. Assume that this claim is true. What is the probability that in a random sample of 400 registered voters taken from this city, less than 49% will favor Maureen Webster?

  8. Answer • Z=(0.49-0.53)/0.02495 = -1.60 • Now, we have P(Z<-1.60) = 0.0548 • Now, let’s look at problem #5.15 a • P(-0.69 <Z<0.69)=0.7549-0.2451=0.5098

  9. 5.2 Sampling Distribution of the Sample Mean • Sample mean versus population mean • Sample standard deviation versus population standard deviation • If X is Normal, then x is normal. • What happens if X is not normal? See applet

  10. Central Limit Theorem • If n is large enough, then a SRS of size n from any population with mean m, and standard deviation s will have the following sampling distribution: x ~N(m, ) • Examples #5.51,5.53 • #5.53 • L=133.225

  11. Example • In the journal Knowledge Quest (Jan/Feb 2002), education professors at the University of Southern California investigated children’s attitudes toward reading. One study measured third through sixth graders’ attitudes toward recreational reading on a 140-point scale. The mean score for this population of children was 106 with a standard deviation of 16.4. In a random sample of 36 children from this population, find P(x<100).

  12. Answer • Z=-2.20 • 0.0139

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