1. Chapter 5 Sampling Distributions

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

3. Notation Sample proportion: = (# in group)/(total) Sample proportion: = (# in group)/(total) Population proportion: p Population proportion: p Example: At UNCW, p=0.45 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? 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 = 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: 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 is approximately normal with mean p and standard deviation Why are we interested in this? Why are we interested in this? Because we can standardize values of and use tables to find probabilities. Because we can standardize values of and use tables to find probabilities.

5. Standardizing sample proportions Z = (observation – mean)/(standard deviation) 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. 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=(0.35-0.33)/0.01662 = 1.2 Z=(0.37-0.33)/0.01662=2.41 Z=(0.37-0.33)/0.01662=2.41 Now we have P(1.2<Z<2.41)=0.992- 0.8849 =.1071 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? 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 Z=(0.49-0.53)/0.02495 = -1.60 Now, we have P(Z<-1.60) = 0.0548 Now, we have P(Z<-1.60) = 0.0548 Now, let’s look at problem #5.15 a Now, let’s look at problem #5.15 a P(-0.69 <Z<0.69)=0.7549-0.2451=0.5098 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 mean versus population mean Sample standard deviation versus population standard deviation Sample standard deviation versus population standard deviation If X is Normal, then  x is normal. If X is Normal, then  x is normal. What happens if X is not normal? See applet 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 , and standard deviation  will have the following sampling distribution: If n is large enough, then a SRS of size n from any population with mean , and standard deviation  will have the following sampling distribution:  x ~N( , ) Examples #5.51,5.53 Examples #5.51,5.53 #5.53 #5.53 a. 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). 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 Z=-2.20 0.0139 0.0139