Statistics Class 15 3/21/2012

Transcription

1 Statistics Class 15 3/21/2012

2 Quiz 1. Cans of regular Pepsi are labeled to indicate that they contain 12 oz. Data Set 17 in Appendix B lists measured amounts for a sample of Pepsi cans. The same statistics are n=36 and x = oz. If the Pepsi cans are filled so that μ = (as labeled) and the population standard deviation is σ = 0.09 oz (based on sample results), find the probability that a sample of 36 cans will have a mean of oz or greater. Do these results suggest that the Pepsi can are filled with an amount greater than oz?

3 We are going to learn ways to estimate the population proportion using a sample proportion We are going to use the sample proportion as our point estimate of the population proportion. We are then going to construct confidence intervals, to estimate the true value of a population proportion. We are going to learn how to interpret confidence intervals. We will discover how to find the sample size necessary to estimate a population proportion.

4 First step is back to the normal standard distribution. We are going to let z α denote the z score with an area of α to its right.

5 First step is back to the normal standard distribution. We are going to let z α denote the z score with an area of α to its right. Lets find z α, where α = 0.025, that is find z.025

6 First step is back to the normal standard distribution. We are going to let z α denote the z score with an area of α to its right. Lets find z α, where α = 0.025, that is find z.025 z.025 = 1. 96

7 A point estimate is a singe value (or point) used to approximate a population parameter.

8 A point estimate is a singe value (or point) used to approximate a population parameter. The sample proportion p is the best point estimate of the population proportion p.

9 A point estimate is a singe value (or point) used to approximate a population parameter. The sample proportion p is the best point estimate of the population proportion p. Lets do example 1!!

10 A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.

11 A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI. A confidence level is the probability 1-α (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. We usually choose confidence levels 90% (α = 0.1), 95%(α = 0.05), and 99%(α = 0.01).

12 From the chapter problem and example 1 The 0.95 confidence interval estimate of the population proportion p is < p <

13 From the chapter problem and example 1 The 0.95 confidence interval estimate of the population proportion p is < p < Correct Interpretation: We are 95% confident that the interval from to actually does contain the true value of the population proportion p. Incorrect Interpretation: There is a 95% chance that the true value of p will fall between and It would also be incorrect to say that 95% of sample proportions fall between and

14 Critical Values A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z α/2 is a critical value that is a z score with the property that it separates an area of α/2 in the right tail of the standard normal distribution. Now lets do example 2!

15 When data from a simple random sample are used to estimate a population proportion, the margin of error, denoted E, is the maximum likely difference (with probability 1-α, such as 0.95) between the observed sample proportion p and the true value of the population proportion p. The margin of error E is also called the maximum error of the estimate and can be found by multiplying the critical value and the standard deviation of sample proportions, as given below: E = z α/2 p q n

16 Now we can construct a Confidence interval for p. 1. Verify the requirements are satisfied (simple random sample, binomial distribution with at least 5 successes and at least 5failures).

17 Now we can construct a Confidence interval for p. 1. Verify the requirements are satisfied (simple random sample, binomial distribution with at least 5 successes and at least 5failures). 2. Find the critical value z α/2, that corresponds to the desired confidence level.

18 Now we can construct a Confidence interval for p. 1. Verify the requirements are satisfied (simple random sample, binomial distribution with at least 5 successes and at least 5failures). 2. Find the critical value z α/2, that corresponds to the desired confidence level. 3. Evaluate the margin of error E = z α/2 pq n

19 Now we can construct a Confidence interval for p. 1. Verify the requirements are satisfied (simple random sample, binomial distribution with at least 5 successes and at least 5failures). 2. Find the critical value z α/2, that corresponds to the desired confidence level. 3. Evaluate the margin of error E = z α/2 pq 4. Using the value of the calculated margin of error E and the value of sample proportion p, find the values of the confidence interval limits p E and p + E. To get (p E, p + E). n

20 Now we can construct a Confidence interval for p. 1. Verify the requirements are satisfied (simple random sample, binomial distribution with at least 5 successes and at least 5failures). 2. Find the critical value z α/2, that corresponds to the desired confidence level. 3. Evaluate the margin of error E = z α/2 pq 4. Using the value of the calculated margin of error E and the value of sample proportion p, find the values of the confidence interval limits p E and p + E. To get (p E, p + E). 5. Round off Confidence interval to 3 significant digits n Now lets do example 3!

21 Determining Sample Size How to determine how large a same should be in order to estimate the population proportion p. If we have an estimate p : n = [z a/2] 2 pq E 2 If we do not have an estimate of p : Taking p = 0.5 and q = 0.5 n = [z a/2] E 2 Round to the next largest whole number.

22 Lets do example 4!

23 Lets do example 4! Finding the Point Estimate and E from a confidence interval p = E = upper confidence limit + (lower confidence limit) 2 upper confidence limit (lower confidence limit) 2 Now do example 5!

24 Homework! 7-2: 1-15odd, and 25-37odd.