Inference for Proportions, Variance and Standard Deviation

Transcription

1 Inference for Proportions, Variance and Standard Deviation Sections 7.10 & 7.6 Cathy Poliak, Ph.D. Office Fleming 11c Department of Mathematics University of Houston Lecture 12 Cathy Poliak, Ph.D. Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 1 / 32

2 Outline 1 Confidence Intervals for Proportions 2 Calculating CI for p 3 Sample Sizes 4 Hypothesis Tests for Proportions 5 Chi-squared Distribution 6 Confidence Intervals for Variance and Standard Deviation Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 2 / 32

3 Confidence Intervals for Proportions Before any inferences can be made about a proportions, certain conditions must be satisfied: 1. The sample must be an SRS from the population of interest. 2. The population must be at least 10 times the size of the sample. 3. The number of successes and number of failures must each be at least 10 (both nˆp 10 and n(1 ˆp) 10). Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 3 / 32

4 Can we use the Confidence Interval? 1. We want to determine the proportion of M&Ms that are brown. In which of the following situations can we create a confidence interval. a) We take a sample of 25 and found 8 to be brown. b) We look at one bag approximately to have 50 M&Ms and notice that 14 are brown. c) We want to determine the proportion of brown M&Ms in one small bag (approximately 50 M&Ms), we take a simple random sample of 25 from the bag a found 8 to be brown. d) We randomly select 100 M&Ms from the entire produced M&Ms and found 25 to be brown. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 4 / 32

5 Example Presidential Approval Rating Suppose a random sample of 1000 Americans were polled and asked if they approve at how the the President is doing his job. 50% of the 1000 said they approve. We want to conduct a 95% confidence interval for the proportion of all Americans that approve of how the President is doing his job. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 5 / 32

6 Change Confidence Level Suppose we get a 95% confidence interval for the proportion of blue M&Ms to be (0.165, 0.335). Suppose we change the confidence level to 92%, what is the new confidence interval? a) (0.15, 0.32) b) (0.17, 0.32) c) (0.17, 0.34) d) (0.165, 0.335) Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 6 / 32

7 Choosing Sample Size You can have both a high confidence while at the same time a small margin of error by taking enough observations. Sample size for confidence intervals of means. Sample size for confidence intervals of proportions. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 7 / 32

8 Example of Determining Sample Size Mars Inc. claims that they produce M&Ms with the following distributions: Brown 30% Red 20% Yellow 20% Orange 10% Green 10% Blue 10% How many M&Ms must be sampled to construct the 98% confidence interval for the proportion of red M&Ms in that bag if we want a margin of error of ±.15? Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 8 / 32

9 Another Example Suppose that prior to conducting a coin-flipping experiment, we suspect that the coin is fair. How many times would we have to flip the coin in order to obtain a 96.5% confidence interval of width of at most.16 for the probability of flipping a head? Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston ) Lecture 12 9 / 32

10 Steps of a Significance Test When performing a significance test, we follow these steps: 1. Check assumptions. 2. State the null and alternative hypothesis. 3. Graph the rejection region, labeling the critical values. 4. Calculate the test statistic. 5. Find the p-value. If this answer is less than the significance level, α, we can reject the null hypothesis in favor of the alternative hypothesis. 6. Give your conclusion using the context of the problem. When stating the conclusion give results with a confidence of (1 α)(100)%. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

11 What if we are not given α? If the P-value for testing H 0 is less than: 0.1 we have some evidence that H 0 is false we have strong evidence that H 0 is false we have very strong evidence that H 0 is false we have extremely strong evidence that H 0 is false. If the P-value is greater than 0.1, we do not have any evidence that H 0 is false. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

12 Inference for a Population Proportion For these inferences, p 0 represents the given population proportion and the hypothesis will be H0 : p = p 0 H a : p p 0 or p < p 0 or p > p 0 Conditions: 1. The sample must be an SRS from the population of interest. 2. The population must be at least 10 times the size of the sample. 3. The number of successes and the number of failures must each be at least 10 (both nˆp 10 and n(1 ˆp) 10). Recall, the statistic used for proportions is: ˆp = # of successes # of observations = x n. For tests involving proportions that meet the above conditions, we will use the z-test statistic: z = ˆp p 0 p 0 (1 p 0 ) n Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

13 Are 10% of M&Ms Blue? A bag of M&Ms was randomly selected from the grocery store shelf, and the color counts were: Brown 14 Red 14 Yellow 5 Orange 7 Green 6 Blue 10 Test if the proportion of M&Ms that are blue are 10%. Use α = Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

14 Example A new shampoo is being test-marketed. A large number of 16-ounce bottles were mailed out at random to potential customers in the hope that the customers will return an enclosed questionnaire. Out of the 1,000 returned questionnaires, 575 indicated that they like the shampoo and will consider buying it when it becomes available on the market. Perform a hypothesis test to determine if the proportion of potential customers is more than 50%. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

15 Example A new brand of chocolate bar is being market tested. Five hundred of the new chocolate bars were given to consumers to try. The consumers were asked whether they liked or disliked the chocolate bar. The company that produces the new brand of chocolate bars said they will put the chocolate bar on the shelf if more than half of consumers (50%) like this chocolate bar. From the sample of 500, 265 people liked the candy bar. Test the claim that more than half of consumers will like this candy bar at the level α = Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

16 Cathy Poliak, Ph.D. Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

17 Which Type of Test? Given the following cases determine which type of test we will use. a) Z-test for mean b) T-test for mean c) Z-test for proportion 1. Hippocrates magazine states that 35 percent of all Americans take multiple vitamins regularly. Suppose a researcher surveyed 750 people to test this claim and found that 296 did regularly take a multiple vitamin. Is this sufficient evidence to conclude that the actual percentage is different from 35% at the 5% significance level? 2. Consumer Reports (January 1993) stated that the mean retail cost of an AT&T model 3730 cellular phone was $600. A random sample of 10 stores in Los Angeles had a mean cost of $ with standard deviation of $ Does this indicate that the mean cost in Los Angeles is less than $600? 3. Athabasca Fishing Lodge is located on Lake Athabasca in northern Canada. In one of their recent brochures, the lodge advertises that 75% of their guests catch northern pike over 20 pounds. Suppose last summer 64 out of a random sample of 83 guests did in fact catch northern pikes weighing over 20 pounds. Does this indicate that the population proportion of guests who catch pikes over 20 pounds is different from 75%? Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

18 Special Gamma Distribution Let ν be a positive number. Then a random variable X is said to have a chi-squared distribution with parameter ν if the pdf of X is the gamma density with α = ν/2 and β = 2. The pdf of a chi-squared random variable is; f (x) = { 1 2 ν/2 Γ(ν/2) x (ν/2) 1 e x/2 x 0 0 x < 0 The parameter ν is called the number of degrees of freedom (df) of X. If X has a chi-squared distribution we often represent this as X χ 2 (ν). Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

19 Plot of Chi-Square Distribution Cathy Poliak, Ph.D. Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

20 Percentiles and Probabilities of Chi-square We can use a chi-square table (pp in text) or pchisq(x,ν) for probability and qchisq(x,ν) for quantiles in R. 1. Determine the 95th percentile of the chis-squared distribution with ν = Determine the 5th percentile of the chi-squared distribution with ν = Determine P(10.98 χ ) with ν = Determine P(χ 2 < or χ 2 > ) with ν = 25. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

21 R code > qchisq(.95,10) [1] > qchisq(.05,10) [1] > pchisq(36.78,22)-pchisq(10.98,22) [1] > pchisq(14.611,25)+ 1 - pchisq(37.652,25) [1] Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

22 Popper #17 Questions 4. Find the values of c 1 and c 2 such that the P(c 1 χ 2 c 2 ) = Use ν = 8. a) c 1 = 1.96, c 2 = 1.96 b) c 1 = , c 2 = c) c 1 = 2.733, c 2 = d) c 1 = 2.180, c 2 = Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

23 Distribution of Z 2 If Z has a standard Normal distribution and X = Z 2, then the pdf of X is; 1 f (x) = 2 1/2 Γ(1/2) x (1/2) 1 e x/2 where x > 0 and f (x) = 0 if x 0. That is X χ 2 (1). Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

24 Proof If x > 0 P(X x) = P(Z 2 x) = P( x Z x) = 2P(0 Z x) = 2 2π x f (x) = d dx = 0 2 x 2π 0 e u2 /2 du e u2 /2 du 1 2 1/2 Γ(1/2) x (1/2) 1 e x/2 Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

25 Theorem 6.7 If X 1 Gamma(α 1, β) and X 2 Gamma(α 2, β) are independent, then X 1 + X 2 Gamma(α 1 + α 2, β). Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

26 Corollaries that Follow from Theorem If X 1, X 2,..., X n are independent and each X i χ 2 (ν 1 ), then X 1 + X X n χ 2 (ν 1 + ν ν n ). 2. If Z 1, Z 2,..., Z n are independent and each has the standard normal distribution, then Z Z Z 2 n χ 2 (n). 3. If X 1, X 2,..., X n are a random sample from a Normal distribution, then X and S 2 are independent. 4. If X 3 = X 1 + X 2 with X 1 χ 2 (ν 1 ), X 3 χ 2 (ν 3 ), ν 3 > ν 1, and X 1 and X 2 are independent, then X 2 χ 2 (ν 3 ν 1 ). 5. If X 1, X 2,..., X n are a random sample from a normal distribution, then (n 1)S2 σ 2 χ 2 (n 1). Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

27 Recall Distribution for Variance (n 1)S 2 σ 2 χ 2 n 1 Thus: P(χ 2 1 α/2,n 1 < (n 1)S2 σ 2 < χ 2 α/2,n 1 ) = 1 α Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

28 Confidence Interval for σ 2 A 100(1 α)% confidence interval for the variance σ 2 of a normal population has lower limit (n 1)s 2 χ 2 α/2,n 1 and upper limit (n 1)s 2 χ 2 1 α/2,n 1 A confidence interval for σ has lower and upper limits that are the square roots of the corresponding limits in the interval for σ 2, where α/2 is the area in the upper tail of the chi-squre distribution. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

29 Example A coffee machine dispenses coffee into paper cups. Here are the amounts measured in a random sample of 20 cups. 9.9, 9.7, 10.0, 10.1, 9.9, 9.6, 9.8, 9.8, 10.0, 9.5, 9.7, 10.1, 9.9, 9.6, 10.2, 9.8, 10.0, 9.9, 9.5, 9.9 Determine a 95% confidence interval for the mean amount of coffee dispensed from this machine. Determine a 95% confidence interval for the standard deviation of the amount of coffee dispensed from this machine. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

30 Cathy Poliak, Ph.D. Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

31 R Code > lcl=(length(coffee)-1)*var(coffee)/qchisq(0.005,(length(coffee)-1),lower > ucl=(length(coffee)-1)*var(coffee)/qchisq(0.005,(length(coffee)-1)) > c(lcl,ucl) [1] > sqrt(c(lcl,ucl)) [1] Cathy Poliak, Ph.D. Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32

32 Example College Placement Tests A random sample of 20 students obtained a mean of x = 72 and a variance of s 2 = 16 on a college placement test in mathematics. Assuming the scores to be normally distributed, construct a 98% confidence interval for the variance. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Sections (Department 7.10 & of 7.6 Mathematics University of Houston Lecture ) / 32