Sampling Distributions and Inference Department of Mathematics & - - PowerPoint PPT Presentation

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 (a) Find µ¯

x

(b) Find P( ¯ X ≥ 28). Find zα/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? (a) ¯ x ± 2.575(σ/√n) (b) ¯ x ± 1.645(σ/√n) (c) ¯ x ± 0.99(σ/√n) 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 t0 such that, (a) P(t ≥ t0) = 0.025, df = 10. (b) P(t ≤ t0) = 0.05, df = 13 (c) P(−t0 < t < t0) = 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 σ2, 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

• f the test.

b) Test the null hypothesis that µ = 100 against the alternative hypothesis that µ = 100, using α = .05. Interpret the results

• f the test.

2 In a test of H0 : µ = 100 versus Ha : 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 Ha : 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 H0 : µ = 6, against H0 : 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 H0 : p = 0.65 against Ha : 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: ¯ x1 = 5, 275, s1 = 150, ¯ x2 = 5, 240, s2 = 200.

a) Use a 95% C. I. to estimate µ1 − µ2. b) Test H0 : µ1 − µ2 = 0 against Ha : µ1 − µ2 = 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: n1 = 17, ¯ x1 = 5.4, s1 = 3.4, n2 = 12, ¯ x2 = 7.9, s2 = 4.8. Assuming equal variances, find a 95% C. I. for µ1 −µ2. Test H0 : µ1 −µ2 = 0 against Ha : µ1 −µ2 = 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 H0 : µd = 0 against Ha : µd = 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 H0 : (p1 − p2) = 0 against H0 : (p1 − p2) < 0. Use α = 0.01 b) Form a 99% C. I. (p1 − p2).

2 Random samples of size n1 = 50 and n2 = 60 were drawn

from populations 1 and 2, respectively. The samples yielded ˆ p1 = 0.4 and ˆ p2 = 0.2. Test H0 : (p1 − p2) = 0.1 against H0 : (p1 − p2) > 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 p1 = .25, p2 = .25, and p3 = .50? Test using α = 0.05. Cell 1 2 3 ni 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

• f 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.

x 5 3

• 1

2 7 6 4 y 4 3 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

Examples: Linear Regresssion

1 Researchers studying a new method for blood typing applied

blood drops to lost cost paper and recorded the rate of absorp- tion ( called blood wicking). The data is shown in the table below. Length (y) 22.5 16 13.5 14 13.75 12.5 Concentration (x) 0.2 0.4 0.6 0.8 1

a) Determine the equation of the straight line model relating y to x. b) Compute the coefficient of correlation and determination for the least squares line.

Memorial University of Newfoundland, October 15, 2016

Examples: Linear Regresssion

1 Consider the following pairs of observations:

y 4 2 5 3 2 4 x 1 4 5 3 2 4

a) Construct a scatterplot of these data. b) Use the method of least squares to fit a straight line to the six points. c) Is there evidence in the data to suggest that x and y are linearly related? Test at α = 0.05 d) Construct a 95% confidence interval for the slope. e) Compute the coefficient of correlation and determination for the least squares line. f) Test for linear correlation in the data at α = 0.01.

2 Describe the slope of the least squares line if

A) r = 0.7 B) r = 0 C) r = −0.1 D) r = 1. What information does the values provide about the relationship between x and y?

Memorial University of Newfoundland, October 15, 2016

Examples: Linear Regresssion

1 Consider the following pairs of measurements:

x 1 2 3 4 5 6 7 y 3 5 4 6 7 7 10

a) Find s2 b) Use a 90% confidence interval to estimate the mean value of y when x = 4. c) Find a 90% prediction interval for a new value of y when x = 4. d) Compare the widths of the intervals in b and c. Which is wider and why?

Memorial University of Newfoundland, October 15, 2016

Thank you!

Memorial University of Newfoundland, October 15, 2016