1. Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 1 Welcome to Unit 8!

2. Section 8.1 Sampling Distributions Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 2 Page 334

3. Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 3 Statements about a large population, but obviously not everyone in the population was surveyed Inferential Statistics are used to draw general conclusions about a population First two statements estimate the mean of a quantity Second two statements estimate a proportion of a population

4. Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 4 Sampling Distributions Sampling Distributions: A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population. Distribution of Sample Means: The sampling distribution that results when we find the means of all possible samples of a given size. The larger the sample size, the more closely it approximates the normal distribution. In all cases, the mean of the distribution of sample means equals the population mean. Distribution of Sample Proportions: The distribution that results when we find the proportions in all possible samples of a given size. The larger the sample size, the more closely it approximates the normal distribution. Sampling Error: The discrepancy between the statistic obtained from the sample and the parameter for the population from which the sample was obtained.

5. Results from a survey of students who were asked how many hours they spend per week using a search engine on the Internet. n = 400 μ = 3.88σ = 2.40 Page 337

6. A sample of 32 students selected from the 400 on the previous slide. The mean of this sample is x = 4.17. 1.1 7.8 6.8 4.9 3.0 6.5 5.2 2.2 5.1 3.4 4.7 7.0 3.8 5.7 6.5 2.7 2.6 1.4 7.1 5.5 3.1 5.0 6.8 6.5 1.7 2.1 1.2 0.3 0.9 2.4 2.5 7.8 Sample 1 ¯ x A different sample of 32 students selected from the 400. Now you have two sample means that don’t agree with each other, and neither one agrees with the true population mean. 1.8 0.4 4.0 2.4 0.8 6.2 0.8 6.6 5.7 7.9 2.5 3.6 5.2 5.7 6.5 1.2 5.4 5.7 7.2 5.1 3.2 3.1 5.0 3.1 0.5 3.9 3.1 5.8 2.9 7.2 0.9 4.0 Sample 2 For this sample is = 3.98. ¯ x

7. Figure 8.6 shows a histogram that results from 100 different samples, each with 32 students. Notice that this histogram is very close to a normal distribution and its mean is very close to the population mean, μ = 3.88. Figure 8.6 A distribution of 100 sample means, with a sample size of n = 32, appears close to a normal distribution with a mean of 3.88.

8. The distribution of sample means is approximately a normal distribution. The mean of the distribution of sample means is 3.88 (the mean of the population). The standard deviation of the distribution of sample means depends on the population standard deviation and the sample size. The population standard deviation is σ = 2.40 and the sample size is n = 32, so the standard deviation of sample means is = = 0.42 σnσn 2.40 32 Central Limit Theorem application: If we were to include all possible samples of size n = 32, this distribution would have these characteristics:

9. Sample Proportions Page 340

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13. Section 8.2 Estimating Population Means Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 13 Page 346

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15. Page 348

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18. Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 18 Determine Minimum Sample Size Solve the margin of error formula [E =2s/√n] for n. You want to study housing costs in the country by sampling recent house sales in various (representative) regions. Your goal is to provide a 95% confidence interval estimate of the housing cost. Previous studies suggest that the population standard deviation is about $7,200. What sample size (at a minimum) should be used to ensure that the sample mean is within a. $500 of the true population mean? E E

19. EXAMPLE Constructing a Confidence Interval You want to study housing costs in the country by sampling recent house sales in various (representative) regions. Your goal is to provide a 95% confidence interval estimate of the housing cost. Previous studies suggest that the population standard deviation is about $7,200. What sample size (at a minimum) should be used to ensure that the sample mean is within $100 of the true population mean?

20. Solution: With E = $100 and σ = $7,200, the minimum sample size that meets the requirements is EXAMPLE Constructing a Confidence Interval E Notice that to decrease the margin of error by a factor of 5 (from $500 to $100), we must increase the sample size by a factor of 25. That is why achieving greater accuracy generally comes with a high cost.

21. Section 8.3 Estimating Population Proportions Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 21 Page 355

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23. The Nielsen ratings for television use a random sample of households. A Nielsen survey results in an estimate that a women’s World Cup soccer game had 72.3% of the entire viewing audience. Assuming that the sample consists of n = 5,000 randomly selected households, find the margin of error and the 95% confidence interval for this estimate. Solution: The sample proportion, = 72.3% = 0.723, is the best estimate of the population proportion. The margin of error is EXAMPLE TV Nielsen Ratings ˆ p

24. Solution: (cont.) The 95% confidence interval is 0.723 – 0.013 < p < 0.723 + 0.013, or With 95% confidence, we conclude that between 71.0% and 73.6% of the entire viewing audience watched the women’s World Cup soccer game. EXAMPLE 2 TV Nielsen Ratings 0.710 < p < 0.736

25. Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 25 You plan a survey to estimate the proportion of students on your campus who carry a cell phone regularly. How many students should be in the sample if you want (with 95% confidence) a margin of error of no more than 4 percentage points? EXAMPLE Minimum Sample Size for Survey

26. Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 26 Solution: Note that 4 percentage points means a margin of error of 0.04. From the given formula, the minimum sample size is You should survey at least 625 students. 1 E 2 n = = = 625 1 0.04 2

27. Core Logic of Hypothesis Testing Considers the probability that the result of a study could have come about if the experimental procedure had no effect If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported

28. Hypothesis Testing using Confidence Intervals  State the claim about the population mean  Determine desired confidence level  Select a random sample from the population  Calculate the confidence interval for the desired level of confidence.  If the claim is contained within the interval, the claim is reasonable; if it is not within the interval, the claim is not reasonable, at the given level of confidence.

29. Unit 9 has no new material! So, the seminar next week will review topics and answer questions. Please email me any topics you would like me to review! EVertuli@kaplan.eduEVertuli@kaplan.edu Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 29