Chapter 10 Two-Sample Inference

Transcription

1 Chapter 10 Two-Sample Inference Independent Samples and Dependent Samples o Two samples are independent when the subjects selected for the first sample do not determine the subjects in the second sample. Two samples are dependent when the subjects in the first sample determine that subjects in the second sample. The data from dependent samples are called matchedpair or paired samples. Confidence Interval for Population Mean Difference (Dependent Samples) o Suppose we have a set of matched-pair data obtained by taking dependent random samples of two populations and finding the differences to produce a random sample of the difference between the populations. A ( ) confidence interval for, the population mean of the differences, is given by lower bound ( ) upper bound ( ) Example 10.1 where and represent the sample mean and sample standard deviation of the differences, respectively, of the set of n paired differences, d 1, d 2, d 3,, d n, and where is based on n 1 degrees of freedom. This t interval applies whenever either of the following condition is met: Case 1: the population of difference is normal, or Case 2: the sample size of difference is large (n 30). The ( ) confidence interval for may also be expressed in the form: ( ) Student Ashley Brittany Chris Dave Emily Fran Greg After (sample 1) Before (sample 2) Q1. Construct a 95% confidence interval for the mean of the differences in the statistics quiz scores. Is there evidence that the Math Center tutoring leads to a mean improvement in the quiz scores? A1. We ignore the original raw data and concentrate only on the set of sample differences: {16, 13, 14, 18, 14, 11, 12} s d (x n x) 2 1 (16 14) 2 (13 14) 2 (13 14) 2 (18 14) 2 (14 14) 2 (11 14) 2 (12 14) For 95% confidence with n 1 6 degrees of freedom, t a/ lower bound x d ta( s d 2 n ) 14 (2.447)(2.3805/7) upper bound x d ta( s d 2 n ) 14 + (2.447)(2.3805/7) We are 95% confident that the population mean of the differences between quiz scores before and after visiting the Math Center lies between points and points.

2 Paired Sample t Test for the Population Mean of the Difference μ d : p-value Method o Suppose we have a set of matched-pair data obtained by taking dependent random sample of two populations and finding the differences to produce a random sample of the difference between the populations. We can use the t test whenever either of the following conditions is met: Case 1: the population of difference is normal, or Case 2: the sample size of difference is large (n 30). Step 1 State the hypotheses and the rejection rule. Use one of the hypothesis test forms from Table Null hypothesis Alternative hypothesis Type of test H 0 : μ d 0 H a : μ d > 0 Right-tailed test H 0 : μ d 0 H a : μ d < 0 Left-tailed test H 0 : μ d 0 H a : μ d 0 Two-tailed test Table 10.1 Forms of the hypothesis test Step 2 Find t data. / Step 3 Find the p-value. Type of hypothesis Test p-values is tail area associated with t data Right-tailed test H 0 : μ d 0 versus H a : μ d > 0 p-value P(t > t data) p-value Area to right of t data 0 t data Left-tailed test H 0 : μ d 0 versus H a : μ d < 0 p-value P(t < t data) p-value Area to left of t data t data 0 Two-tailed test H 0 : μ d 0 versus H a : μ d 0 p-value P(t > ) + P(t < - ) 2 * P(t > ) Sum of the two tail areas. Sum of two areas is p-value t data t data Step 4 State the conclusion and interpretation. Compare the p-value with 0

3 Example 10.2 Q1. Paired-sample t test for μ d : The p-value method A1. pg.549 Paired Sample t Test for the Population Mean of the Difference μ d : Critical Value Method o Suppose we have a set of matched-pair data obtained by taking dependent random sample of two populations and finding the differences to produce a random sample of the difference between the populations. You can use the t test whenever either of the following conditions is met: Case 1: the population of difference is normal, or Case 2: the sample size of difference is large (n 30). Step 1 State the hypotheses. Use one of the hypothesis test forms from Table State clearly the meaning of μ d. Step 2 Find t crit, and state the rejection rule. To find t crit, use the t table and degrees of freedom n 1. To find the rejection rule, use Table Form of test Rejection rules: Reject H 0 if Right-tailed H 0 : μ d 0 vs. H a : μ d > 0 t data > t crit Left-tailed H 0 : μ d 0 vs. H a : μ d < 0 t data < t crit Tow-tailed H 0 : μ d 0 vs. H a : μ d 0 t data > t crit or t data < t crit Table 10.2 Rejection rules for the t test for μ d Step 3 Find t data. / Step 4 State the conclusion and interpretation. Compare the t data with t crit. Example 10.3 Q1. Paired-sample t test for μ d :The critical value method A1. pg.551

4 Sampling Distribution of 1 2 o When random samples are drawn independently from two populations with population means μ 1 and μ 2, and either Case 1: the two populations are normally distributed, or Case 2: the two sample sizes are large (at least 30), then the quantity ( ) ( ) ( ) ( ) approximately follows a t distribution with degrees of freedom equal to the smaller of n 1 1 and n 2 1, where and s 1 represent the mean and standard deviation of the sample taken from population 1, and and s 2 represent the mean and standard deviation of the sample taken from population 2. Standard Error of 1 2 o The standard error of the statistic 1 2 is It measures the size of the typical error in using 1 2 to estimate. Confidence Interval for o For two independent random samples taken from two populations with population means and, and 100(1 )% confidence interval for is given by ( ) / The t interval applies whenever either of the following conditions is met: Case 1: both populations are normally distributed, or Case 2: both sample sizes are large. Margin of Error E o The margin of error for a 100(1 )% confidence interval for given by E / * (standard error) / * ( ) is / *

5 Example 10.4 Gender Females (sample 1) Males (sample 2) Sample size Sample mean body temperature Sample standard deviation Population mean body temperature n 1 65 x S μ? n 2 65 x S μ? Summary statistics for female versus male body temperatures in 0 F Q1. Calculate the standard error s x x temperature between women and men. for estimating the difference in population mean body A1. s x1 x 2 s 1 2 n 1 s n Q2. Find a 95% confidence interval for the difference in women s and men s population men body temperatures. A2. Both sample size are large, so the sampling distribution of x 1 x 2 has a t distribution. We know the standard error s x1 x But we need to find t a/2 to use the formula for E. the require degrees of freedom is the smaller of n 1 1 and n 2 1, which are both equal to , so the degrees of freedom for t a/2 is also 64. This df 64 is not listed in the t table, so we choose the next lowest df listed, 60. For 95% confidence, then, t a/2 2.00, and the margin of error is E t a/2 * (s x1 x 2 ) (2.00)*(0.1265) The 95% confidence interval is then (x 1 x 2 ) E ( ) (0. 036, )

6 Sampling Distribution of o When two random samples are drawn independently from two populations, then the quantity ( ) ( ) ( ) ( ) has an approximately standard normal distribution when the following conditions are satisfied: x 1 5, (n 1 x 1 ) 5, x 2 5, (n 2 x 2 ) 5 Standard Error of o The standard error of the statistic is where and. The standard error size of the typical error in using to estimate. measures the Confidence Interval for o For two independent random samples taken from two populations with population proportion and, a 100(1 )% confidence interval for is given by / Margin of Error E o The margin of error for a 100(1 )% confidence interval for given by is / ( ) / ( ) /

7 Example 10.5 Boys girls Number responding Yes x x 2 93 Sample size n n Sample proportion p1 x 1 / n 1 195/ p2 x 2 / n 2 93/ Proportions of teenage boys and girls who post their last names in online profiles Q1. Find the point estimate p p for the difference in population proportions p p Q2. Calculate the standard error p1 p q1 1 p and q2 1 p s p1 p2 p1q1 n 1 p2q2 n 2 ( )( ) 487 ( )( ) Q3. For a 95% confidence level, calculate the margin of error. E Z a/2 (s p1 p2) 1. 96( ) Q4. Construct and interpret a 95% confidence interval for the difference in population proportions of girls and boys whose last name is posted to their online profile. p1 p2 E ( ) ( , )