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This chapter covers the concepts of estimation and hypothesis testing for large sample populations. It includes point and interval estimators, properties of good estimators, and large sample test statistics using the z-distribution.
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Statistikbagisainsgunaan MTH3003 PJJ SEM II 2014/2015 F2F II 12/4/2015
Assessment: • ASSIGNMENT :25% Assignment 1 (10%) Assignment 2 (15%) • Mid exam :30% Part A (Objective) Part B (Subjective) • Final Exam: 40% Part A (Objective) Part B (Subjective - Short) Part C (Subjective – Long)
CHAPTER 8 ESTIMATION (LARGE SAMPLE) • Definition • Types of estimators: • Point estimator • Interval estimator
Key Concepts I. Types of Estimators 1. Point estimator: a single number is calculated to estimate the population parameter. 2. Interval estimator: two numbers are calculated to form an interval that contains the parameter. II. Properties of Good Estimators 1. Unbiased: the average value of the estimator equals the parameter to be estimated. 2. Minimum variance: of all the unbiased estimators, the best estimator has a sampling distribution with the smallest standard error. 3. The margin of error measures the maximum distance between the estimator and the true value of the parameter.
The Margin of Error • Margin of error: The maximum error of estimation, is the maximum likely difference observed between sample mean x and true population mean µ, calculated as : 1.645 1.96 2.33 2.575
Key Concepts III. Large-Sample Point Estimators To estimate one of four population parameters when the sample sizes are large, use the following point estimators with the appropriate margins of error.
Example 1 • A homeowner randomly samples 64 homes similar to her own and finds that the average selling price is $252,000 with a standard deviation of $15,000. Estimate the average selling price for all similar homes in the city.
Example 2 A quality control technician wants to estimate the proportion of soda cans that are underfilled. He randomly samples 200 cans of soda and finds 10 underfilled cans.
Key Concepts IV. Large-Sample Interval Estimators To estimate one of four population parameters when the sample sizes are large, use the following interval estimators.
Example 3 A random sample of n = 50 males showed a mean average daily intake of dairy products equal to 756 grams with a standard deviation of 35 grams. Find a 95% confidence interval for the population average m. 1.96
Example 4 Of a random sample of n = 150 college students, 104 of the students said that they had played on a soccer team during their K-12 years. Estimate the proportion of college students who played soccer in their youth with a 98% confidence interval. 2.33
Example 5 1.96 • Compare the average daily intake of dairy products of men and women using a 95% confidence interval.
Example 6 • Compare the proportion of male and female college students who said that they had played on a soccer team during their K-12 years using a 99% confidence interval. 2.575
CHAPTER 9 LARGE SAMPLE TESTS OF HYPOTHESES PART I Testing the single mean & single proportion PART II Testing the difference between two means & difference between two proportions
Key Concepts I. Parts of a Statistical Test 1. Null hypothesis: a contradiction of the alternative hypothesis 2. Alternative hypothesis: the hypothesis the researcher wants to support. 3.Test statistic and its p-value: sample evidence calculated from sample data. 4. Rejectionregion—critical values and significance levels: values that separate rejection and nonrejection of the null hypothesis 5. Conclusion: Reject or do not reject the null hypothesis, stating the practical significance of your conclusion.
Key Concepts II. Errors and Statistical Significance 1. The significance level a is the probability if rejecting H0 when it is in fact true. 2. The p-valueis the probability of observing a test statistic as extreme as or more than the one observed; also, the smallest value of a for which H0 can be rejected. 3. When the p-valueis less than the significance level a , the null hypothesis is rejected. This happens when the test statistic exceeds the critical value. 4. In a Type II error, b is the probability of accepting H0 when it is in fact false. The power of the test is (1 -b), the probability of rejecting H0 when it is false.
Key Concepts III. Large-Sample Test Statistics Using the z Distribution To test one of the four population parameters when the sample sizes are large, use the following test statistics:
Example 1 (testing the single mean) The daily yield for a chemical plant has averaged 880 tons for several years. The quality control manager wants to know if this average has changed. She randomly selects 50 days and records an average yield of 871 tons with a standard deviation of 21 tons.
Using critical value approach: What is the critical value of z that cuts off exactly a/2 = .01/2 = .005 in the tail of the z distribution? For our example, z = -3.03 falls in the rejection region and H0 is rejected at the 1% significance level. Rejection Region: Reject H0 if z > 2.58 or z < -2.58. If the test statistic falls in the rejection region, its p-value will be less than a = .01.
Using p-value approach: This is an unlikely occurrence, which happens about 2 times in 1000, assuming m = 880!
Example 2 (testing the single proportion) Regardless of age, about 20% of American adults participate in fitness activities at least twice a week. A random sample of 100 adults over 40 years old found 15 who exercised at least twice a week. Is this evidence of a decline in participation after age 40? Use a = .05.
Using critical value approach: What is the critical value of z that cuts off exactly a= .05 in the left-tail of the z distribution? For our example, z = -1.25 does not fall in the rejection region and H0 is not rejected. There is not enough evidence to indicate that p is less than .2 for people over 40. Rejection Region: Reject H0 if z < -1.645. If the test statistic falls in the rejection region, its p-value will be less than a = .05.
Example 3 (testing difference between two means) • Is there a difference in the average daily intakes of dairy products for men versus women? Use a = .05.
Using p-value approach: Since the p-value is greater than a = .05, H0 is not rejected. There is insufficient evidence to indicate that men and women have different average daily intakes.
Example 4 (testing difference between two proportions) Compare the proportion of male and female college students who said that they had played on a soccer team during their K-12 years using a test of hypothesis.
Using p-value approach: Since the p-value is less than a = .01, H0 is rejected. The results are highly significant. There is evidence to indicate that the rates of participation are different for boys and girls.
CHAPTER 10 INFERENCE FROM SMALL SAMPLE PART I Testing the single mean & difference between two means PART II Testing the single variance & ratio of two variances
Key Concepts I. Experimental Designs for Small Samples 1. Single random sample: The sampled population must be normal. 2. Two independent random samples: Both sampled populations must be normal. a. Populations have a common variance s2. b. Populations have different variances 3. Paired-difference or matched-pairs design: The samples are not independent.
Key Concepts II. Statistical Tests of Significance 1. Based on the t, F, and c2 distributions 2. Use the same procedure as in Chapter 9 3. Rejection region—critical values and significance levels: based on the t,F, and c2 distributions with the appropriate degrees of freedom 4. Tests of population parameters: a single mean, the difference between two means, a single variance, and the ratio of two variances III. Small Sample Test Statistics To test one of the population parameters when the sample sizes are small, use the following test statistics:
Testing the single mean m Two-tailed One-tailed (lower-tail) One-tailed (upper-tail) Using p-values or a rejection region based on t distribution with df = n-1 The basic procedures are the same as those used for large samples. For a test of hypothesis:
Confidence Interval For a 100(1-a)% confidence interval for the population mean m:
Example 1 A sprinkler (sprayer) system is designed so that the average time for the sprinklers to activate after being turned on is no more than 15 seconds. A test of 5 systems gave the following times: 17, 31, 12, 17, 13, 25 Is the system working as specified? Test using a = .05.
Solution Data: 17, 31, 12, 17, 13, 25 First, calculate the sample mean and standard deviation, using your calculator or the formulas in Chapter 2.
Rejection Region: Reject H0 if t > 2.015. If the test statistic falls in the rejection region, its p-value will be less than a = .05. Data: 17, 31, 12, 17, 13, 25 Calculate the test statistic and find the rejection region for a =.05.
Conclusion:For our example, t = 1.38 does not fall in the rejection region and H0 is not rejected. There is insufficient evidence to indicate that the average activation time is greater than 15. Data: 17, 31, 12, 17, 13, 25 Compare the observed test statistic to the rejection region, and draw conclusions.
Testing the difference between two means (Independent Samples) • As in Chapter 9, independent random samples of size n1 and n2 are drawn from population 1 and population 2 with means m1danm2,and variances and . • Since the sample sizes are small, the two populations must be normal Two-tailed One-tailed (lower-tail) One-tailed (upper-tail) • The basic procedures are the same as those used for large samples. For a test of hypothesis:
Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30) Interval Estimate with 2 Unknown
Test Statistics ( ) • Instead of estimating each population variance separately, we estimate the common variance with • And the resulting test statistic, has a t distribution with n1+n2-2 degrees of freedom.
Test Statistics (cont’d) • How to check the reasonable equality of variance assumption? Rule of Thumb Assume that the variance are equal Do Not Assume that the variance are equal
Test Statistics ( ) • If the population variances cannot be assumed equal, the test statistic is • It has an approximate t distribution with degrees of freedom of
Confidence Interval ( ) • You can also create a 100(1-a)% confidence interval form1-m2. • Remember the three assumptions: • Original populations normal • Samples random and independent • Equal population variances.
Example 2 Two training procedures are compared by measuring the time that it takes trainees to assemble a device. A different group of trainees are taught using each method. Is there a difference in the two methods? Use a = 0.01.
Solution Hypothesis Equality of Variances Checking Test Statistics
Using critical value approach: What is the critical value of t that cuts off exactly a/2 = .01/2 = .005 in the tail of the t distribution? Critical value: For our example, t = 1.99 falls in the rejection region and H0 is rejected at the 1% significance level. Rejection Region: Reject H0 if t > 2.845 or t < -2.845. If the test statistic falls in the rejection region, its p-value will be less than a = .01
The Paired-Difference Test (dependent samples) • Sometimes the assumption of independent samples is intentionally violated, resulting in amatched-pairsorpaired-difference test. • By designing the experiment in this way, we can eliminate unwanted variability in the experiment by analyzing only the differences, • di = x1i – x2i • to see if there is a difference in the two population means, m1-m2.
Example 3 • One Type A and one Type B tire are randomly assigned to each of the rear wheels of five cars. Compare the average tire wear for types A and B using a test of hypothesis. • But the samples are not independent. The pairs of responses are linked because measurements are taken on the same car.
Rejection region: Reject H0 if |t| > 2.776. Conclusion: Since ttable= 12.8, H0 is rejected. There is a difference in the average tire wear for the two types of tires. Solution
