Sampling Distributions and Inference Department of Mathematics & Statistics Memorial University October 15, 2016 Memorial University of Newfoundland, October 15, 2016
Examples: Sampling Distribution and Confidence Intervals A random sample of size n = 100 observations is selected from a population with µ = 30 and σ = 16 (b) Find P ( ¯ (a) Find µ ¯ X ≥ 28). x Find z α/ 2 if, (a) α = 0 . 2 (b) α = 0 . 01. What is the confidence level of each of the following C. I.s for µ ? x ± 2 . 575( σ/ √ n ) x ± 1 . 645( σ/ √ n ) (a) ¯ (b) ¯ x ± 0 . 99( σ/ √ n ) (c) ¯ A r. s. of n observations was selected from a population with unknown mean µ and standard deviation σ = 20. Calculate a 95% C. I. for µ if n = 75, ¯ x = 28. Memorial University of Newfoundland, October 15, 2016
Examples: Confidence Intervals Suppose n = 7 , 30 , 1000, compare the z-values to the t-values when the confidence level is (a) 80% (b) 98%. Find t 0 such that, (a) P ( t ≥ t 0 ) = 0 . 025, df = 10. (b) P ( t ≤ t 0 ) = 0 . 05, df = 13 (c) P ( − t 0 < t < t 0 ) = 0 . 95, df = 16. The following random sample was selected from a normal distri- bution: 4, 6, 3, 5, 9, 3. Construct a 90% C. I. for the population mean. A r. s. of size n = 196 yielded ˆ p = 0 . 64. (a) Is the sample size large enough to use the normal approxi- mation to construct a C. I. for p ? (b) Construct a 95% C. I. for p . Memorial University of Newfoundland, October 15, 2016
Examples: Determining Sample Size Suppose you wish to estimate a population mean correct to within 0.15 with a confidence level of 0.90. You do not know σ 2 , but you know that the observations will range in value between 31 and 39. Find the approximate sample size that will produce the desired accuracy of the estimate. Find the approximate sample size required to construct a 95% confidence interval for p that has sampling error SE = 0.06 if you have no prior knowledge about p but you wish to be certain that your sample is large enough to achieve the specified accuracy for the estimate. A 90% C.I for p is (.26,.54). How large was the sample size used to construct this interval? Memorial University of Newfoundland, October 15, 2016
Examples: Hypothesis Testing 1 A random sample of 100 observations from a population with standard deviation 60 yielded a sample mean of 110. a) Test the null hypothesis that µ = 100 against the alternative hypothesis that µ > 100, using α = . 05. Interpret the results of the test. b) Test the null hypothesis that µ = 100 against the alternative hypothesis that µ � = 100, using α = . 05. Interpret the results of the test. 2 In a test of H 0 : µ = 100 versus H a : µ � = 100, the sample data yielded the test statistic z = 2 . 17. Find the p-value for the test. What is the p-value if the alternative were H a : µ > 100? Memorial University of Newfoundland, October 15, 2016
Examples: Hypothesis Testing 1 Light bulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected, the lifetime of each bulb determined and the average lifetime was 738.44 hours with a standard deviation of 38.20 hours. Use the p-value to test the hypotheses. 2 A sample of five measurements randomly selected from a nor- mally distributed population, resulted in ¯ x = 4 . 8, and s = 1 . 3. Test H 0 : µ = 6, against H 0 : µ < 6 at α = 0 . 05. Memorial University of Newfoundland, October 15, 2016
Examples: Hypothesis Testing 1 A random sample of 100 observations is selected from a bino- mial population with unknown probability of success, p . The computed value of ˆ p = 0 . 74. Test H 0 : p = 0 . 65 against H a : p > 0 . 65, at α = 0 . 01. 2 In order to compare the means of two populations, two indepen- dent random samples of 400 observations are selected from each population with the following results: ¯ x 1 = 5 , 275, s 1 = 150, ¯ x 2 = 5 , 240, s 2 = 200. a) Use a 95% C. I. to estimate µ 1 − µ 2 . b) Test H 0 : µ 1 − µ 2 = 0 against H a : µ 1 − µ 2 � = 0. Give the p-value of the test. 3 Independent random samples selected from two normal pop- ulations produced the following sample means and standard deviations: n 1 = 17 , ¯ x 1 = 5 . 4 , s 1 = 3 . 4, n 2 = 12 , ¯ x 2 = 7 . 9 , s 2 = 4 . 8. Assuming equal variances, find a 95% C. I. for µ 1 − µ 2 . Test H 0 : µ 1 − µ 2 = 0 against H a : µ 1 − µ 2 � = 0 using α = 0 . 05. Memorial University of Newfoundland, October 15, 2016
Examples: Hypothesis Testing 1 The data for a random sample of six paired observations are shown in the following table. Pair Sample 1 Sample 2 1 7 4 2 3 1 3 9 7 4 6 2 5 4 4 6 8 7 a) Construct a 95% confidence interval for the difference in means. b) Test H 0 : µ d = 0 against H a : µ d � = 0. Use α = 0 . 05. Memorial University of Newfoundland, October 15, 2016
Examples: Hypothesis Testing 1 Camp Jump Start is an 8-week summer camp for overweight and obese adolescents. Counselors develop a weight manage- ment program for each camper that centers on nutrition edu- cation and physical activity. In a study published in Pediatrics (April 2010), the summary statistics for body mass index (BMI) measured for each of 76 campers both at the start and end of camp are shown below Mean Standard Deviation Starting BMI 34.9 6.9 Ending BMI 31.6 6.2 Paired Differences 3.3 1.5 a) Give the null hypothesis for determining whether the mean BMI at the end of camp is less than the mean BMI at the start of camp. test the hypotheses using the p-value. b) Find the 99% C. I. for the true mean change in BMI for Camp Jump Start campers. Memorial University of Newfoundland, October 15, 2016
Examples: Hypothesis Testing 1 Independent random samples, each containing 800 observa- tions, were selected from two binomial populations. The sam- ples from populations 1 and 2 produced 320 and 400 successes, respectively. a) Test H 0 : ( p 1 − p 2 ) = 0 against H 0 : ( p 1 − p 2 ) < 0. Use α = 0 . 01 b) Form a 99% C. I. ( p 1 − p 2 ). 2 Random samples of size n 1 = 50 and n 2 = 60 were drawn from populations 1 and 2, respectively. The samples yielded ˆ p 1 = 0 . 4 and ˆ p 2 = 0 . 2. Test H 0 : ( p 1 − p 2 ) = 0 . 1 against H 0 : ( p 1 − p 2 ) > 0 . 1. Use α = 0 . 01 Memorial University of Newfoundland, October 15, 2016
Examples: Categorical Data 1 A multinomial experiment with k = 3 cells and n = 320 pro- duced the data shown in the accompanying table. Do these data provide sufficient evidence to contradict the null hypothe- sis that p 1 = . 25, p 2 = . 25, and p 3 = . 50? Test using α = 0 . 05. Cell 1 2 3 n i 78 60 182 Memorial University of Newfoundland, October 15, 2016
Examples: Categorical Data 1 Inc. Technology (Mar. 18, 1997) reported the results of an Equifax/Harris Decima Consumer Privacy Survey in which 328 Internet users indicated their level of agreement with the fol- lowing statement: “The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes.” the number of users in each response cate- gory is summarized as follows: Agree Agree Disagree Disagree Strongly Somewhat Somewhat Strongly 59 108 82 79 Does the data provide sufficient evidence that the opinion of Internet users are evenly divided among the four categories. Test at α = 0 . 05. Memorial University of Newfoundland, October 15, 2016
Examples: Categorical Data 1 In Health Education Research (Feb. 2005), nutrition scientists investigated children’s perception of their environments. Each in a sample of 147 ten-year old children drew maps of their home and neighborhood environment. The results broken down by gender, for one theme are shown in the table below Presence of TV Number of Number of in Bedroom Boys Girls Yes 11 9 No 66 61 Total 77 70 Conduct a test to determine whether the likelihood of drawing a TV in the bedroom is different for boys and girls. Use α = 0 . 05. Memorial University of Newfoundland, October 15, 2016
Examples: Categorical Data 1 A study was conducted to examine the political strategies used by ethnic groups worldwide in their fight for minority rights. Each in a sample of 275 ethnic groups was classified according to world region and highest level of political action reported. The data are summarized in a contingency table below. Con- duct a test at α = 0 . 10 to determine whether political strategy of ethnic groups depends on world region. Political Strategy No Mass Rebellion, World Region Action Action Civil War Latin America 24 31 7 Post-Communist 32 23 4 Asia 11 22 26 Africa/Middle East 39 36 20 Memorial University of Newfoundland, October 15, 2016
Examples: Linear Regresssion 1 Consider the following pairs of measurements. 5 3 -1 2 7 6 4 x y 4 3 0 1 8 5 3 a) Construct a scatterplot of these data. b) Estimate the least squares line. c) Plot the least squares line on your scatterplot. Does the line appear to fit the data well? d) Interpret the y -intercept and slope of the least squares line. Over what range of x are these interpretations meaningful? Memorial University of Newfoundland, October 15, 2016
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