1. Section 8.1 Sampling Distributions Page 334 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.

2. 2 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: A sampling distribution of the mean gives all the values the mean can take, along with the probability of getting each value if sampling is random from the null-hypothesis population. Distribution of Sample Proportions: The distribution that results when we find the proportions ( ṕ ) in all possible samples of a given size. Sampling Error: The discrepancy between the statistic obtained from the sample and the parameter for the population from which the sample was obtained.

3. 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

4. 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

5. 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.

6. 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:

7. Sample Proportions In a survey where 400 students were asked if they own a car, 240 replied that they did. The exact proportion of car owners is p = = 0.6 240 400 This population proportion, p = 0.6, is another example of a population parameter. Page 340

8. Section 8.2 Estimating Population Means Page 346 8 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.

9. 9 Margin of Error The margin of error for the 95% confidence interval is where s is the standard deviation of the sample. We find the 95% confidence interval by adding and subtracting the margin of error from the sample mean. That is, the 95% confidence interval ranges from (x – margin of error) to (x + margin of error) We can write this confidence interval more formally as – E < μ < + E or more briefly as ± E margin of error = E ≈ 2s n2s n ¯ x ¯ x ¯ x

10. 95% Confidence Interval Page 348

11. Confidence Intervals in StatCrunch Videos made by other instructors: – Find a Confidence Interval for a population mean using StatCrunch http://screencast.com/t/rTYiEKGo3ww – Find a Confidence Interval for a population proportion using StatCrunch http://screencast.com/t/eISO0FRlQu 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. 11

12. 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. 12

13. 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 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

14. Solution: a. With E = $500 and σ estimated as $7,200, the minimum sample size that meets the requirements is 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 a. $500 of the true population mean? b. $100 of the true population mean? E

15. Solution: a.(cont.) Because the sample size must be a whole number, we conclude that the sample should include at least 830 prices. b.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.

16. Section 8.3 Estimating Population Proportions Page 355 16 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.

17. 17 95% Confidence Interval for a Population Proportion For a population proportion, the margin of error for the 95% confidence interval is where is the sample proportion., and z* stands for critical z score 90% confidence z* = 1.645 95% confidence z* = 1..96 (WARNING! The MSL program requires you to use 2 here) 99% confidence z* = 2.326

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 The 95% confidence interval ( for proportions) ranges from p hat – margin of error to p hat + margin of error We can write this confidence interval more formally as p hat +/- margin of error

19. 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. 19 Choosing the Correct Sample Size In order to estimate a population proportion with a 95% degree of confidence and a specified margin of error of E, the size of the sample should be at least n = 1 E 2

20. 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

21. 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

22. Confidence Interval - Means The confidence interval for a population mean is as follows: x bar – E < μ < x bar + E E is the margin of error, and here it is defined as follows: E = z* (σ / square root of n) (Remember to use 2 for z* in the MSL program!) 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. 22

23. 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. 23 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? 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. EXAMPLE Minimum Sample Size for Survey 1 E 2 n = = = 625 1 0.04 2

24. 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

25. 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.  See Testing a Claim document in Doc Sharing

26. Testing a Claim Example Testing a claim…confidence interval example (for means) A school claims that the mean math ACT scores of their incoming freshman is 20.39, with a standard deviation of 4. A sample of 30 freshmen finds that the average of the same is 19. Test the school’s claim with a 95% confidence interval 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

27. Example Continued First, we compute our confidence interval X bar stands for the mean from the sample, so here x bar = 19.3 Now, we need our margin of error. Again, the margin of error is E = z* (σ / square root of n) So…E = 2 * (4 / sq(30)) = 1.46059 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. 27

28. Example Continued Therefore, since our x bar is 19.3 (the result from the sample) and our margin of error is 1.46059, the 95% confidence interval is: 19.3 – 1.46059 < μ < 19.3 + 1.46059 17.83941 < μ < 20.76059 Now we check to see if the claimed value (20 in this case) is inside the confidence interval. 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. 28

29. Example Continued If it is, we say the claim is reasonable If not, we say the claim is unreasonable Here, our claim of 20 is inside the confidence interval, so we say the claim is reasonable Note though that if the claim had been 22 we would say the claim is unreasonable (why?) 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