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# Sampling Distributions

Sampling Distributions. Chapter 8. Distributions of the Sample Mean. Lesson 8.1. Sampling Statistics.

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## Sampling Distributions

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1. Sampling Distributions Chapter 8

2. Distributions of the Sample Mean Lesson 8.1

3. Sampling Statistics • Since statistics are actually random variables associated with a given sample, they will vary from sample to sample. Therefore, they have probabilities distributions associated with them. This will allows us to find probabilities associated with the sample. • i.e. what is the probability that the mean of the population matches the mean of your sample.

4. Sampling Distributions about the mean • Obtain a simple random sample of size n. • Compute the sample mean. • Repeat steps 1 and 2 until all simple random samples have been obtained from the population.

5. Example The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams. Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n = 5 from this population.

6. The data on the following slide represent the sample means for the 200 simple random samples of size n = 5. • For example, the first sample of n = 5 had the following data: • 2.493 2.466 2.473 2.492 2.471 • Note: = 2.479 for this sample

7. Effect of Sample Size • Repeat Experiment using sample size of n = 20 • The mean of the 200 sample means for n =20 is still 2.46, but the standard deviation is now 0.0045 (0.0086 for n = 5). • As expected, there is less variability in the distribution of the sample mean with n =20 than with n =5.

8. Mean and Standard Deviation of • Suppose a simple random sample of size n is drawn from a large population with mean µ and standard deviation The sampling distribution of will have:

9. Example • Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes • If a random sample of n = 35 oil changes I selected describe the sampling distribution of the sample mean. • If a random sample of n = 35 oil changes is selected, what is the probability that the mean oil change time is less than 11 minutes.

10. #21: The length of human pregnancies is approximately normally distributed with man 266 days and standard deviation of 16 days • What is the probability a randomly selected pregnancy lasts less than 260 days. • Suppose a random sample of 20 pregnancies is obtained. Describe the sampling distribution of the sample mean length of human pregnancies. • What is the probability that a random sample of 20 pregnancies has a mean gestation period of 260 days or less? • What is the probability that a random sample of 50 pregnancies has a mean gestation period of 260 days or less? • What might you conclude if a random sample of 50 pregnancies resulted in a mean gestation period of 260 days or less?

11. Distribution of Sample Proportions Lesson 8.2

12. Point estimate of a population proportion • Number of individuals of a sample with a certain characteristic • Number of people whose favorite color is red • Number of roses in the greenhouse with yellow flowers Sample Proportion (p-hat) 

13. Example • In a Quinnipiac University Poll conducted in May of 2008, 1,745 registered voters nationwide were asked whether they approved of the way George W. Bush is handling the economy. 349 responded “yes”. Obtain a point estimate for the proportion of registered voters who approve of the way George W. Bush is handling the economy.

14. Using Simulation to Describe the Distribution of the Sample Proportion According to a Time poll conducted in June of 2008, 42% of registered voters believed that gay and lesbian couples should be allowed to marry. Describe the sampling distribution of the sample proportion for samples of size n=10, 50, 100.

15. Key Points from Example 2 • Shape: As the size of the sample, n, increases, the shape of the sampling distribution of the sample proportion becomes approximately normal. • Center: The mean of the sampling distribution of the sample proportion equals the population proportion, p. • Spread:The standard deviation of the sampling distribution of the sample proportion decreases as the sample size, n, increases.

16. Sampling distribution characteristics • Shape is approximately normal provided • Mean of the sampling distribution is • Standard deviation of sampling distribution is • Values independent ….. Verify n is no more than 5% of the population

17. According to a Time poll conducted in June of 2008, 42% of registered voters believed that gay and lesbian couples should be allowed to marry. Suppose that we obtain a simple random sample of 50 voters and determine which believe that gay and lesbian couples should be allowed to marry. Describe the sampling distribution of the sample proportion for registered voters who believe that gay and lesbian couples should be allowed to marry.

18. Compute probabilities of sample proportions According to the Centers for Disease Control and Prevention, 18.8% of school-aged children, aged 6-11 years, were overweight in 2004. • In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight? • Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude?

19. #17: according to a USA today Snapshot, 26% of adults do not have any credit cards. A simple random sample of 500 adults is obtained.

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