1 Chapter 9 Inferences from Two Samples In this chapter we will deal with two samples from two populations. The general goal is to compare the parameters of the two populations. For the first population we use index 1, for the second population index 2.

2 Section 9-2 Two Proportions

3 p 2, n 2, x 2, p 2, and q 2 are used for the second population. ^^ Notation for Two Proportions For the first population, we let: p 1 = first population proportion n 1 = size of the first sample x 1 = number of successes in the first sample ^ p 1 = (the first sample proportion) q 1 = 1 – p 1 ^ ^ n1n1 x1x1

4  The pooled sample proportion is denoted by p and is given by: Pooled Sample Proportion = p n 1 + n 2 x 1 + x 2  We denote q = 1 – p

5 Requirements 1. We have two independent random samples. 2. For each of the two samples, the number of successes is at least 5 and the number of failures is at least 5.

6 Tests for Two Proportions H 0 : p 1 = p 2 H 1 : p 1  p 2, H 1 : p 1 p 2 Note: no numerical values for p 1 or p 2 are claimed in the hypotheses. The goal is to compare the two proportions. two tails left tail right tail

7 Test Statistic for Two Proportions + z =z = ( p 1 – p 2 ) – ( p 1 – p 2 ) ^ ^ n1n1 pq n2n2 = p1p1 ^ x1x1 n1n1 p2p2 ^ x2x2 n2n2 = and q = 1 – p n 1 + n 2 p = x 1 + x 2 Note: p 1 – p 2 =0 according to H 0

8 Example: The table below lists results from a simple random sample of front-seat occupants involved in car crashes. Use a 0.05 significance level to test the claim that the fatality rate of occupants is lower for those in cars equipped with airbags.

9 Example: Requirements are satisfied: two simple random samples, two samples are independent; Each has at least 5 successes and 5 failures. Step 1: Express the claim as p 1 < p 2. Step 2: p 1 < p 2 does not contain equality so it is the alternative hypothesis. The null hypothesis is the statement of equality.

10 Example: H 0 : p 1 = p 2 H 1 : p 1 < p 2 (original claim) Step 3: Significance level is 0.05 Step 4: Compute the pooled proportion: With it follows

11 Example: Step 5: Find the value of the test statistic.

12 Example: Left-tailed test. Area to left of z = –1.91 is (Table A-2), so the P-value is

13 Example: Step 6: Because the P-value of is less than the significance level of  = 0.05, we reject the null hypothesis of p 1 = p 2. Because we reject the null hypothesis, we conclude that there is sufficient evidence to support the original claim. Final conclusion: the proportion of accident fatalities for occupants in cars with airbags is less than the proportion of fatalities for occupants in cars without airbags.

14 Example: Using the Traditional Method the critical value is z = – The test statistic of z = –1.91 does fall in the critical region bounded by the critical value of z = – We again reject the null hypothesis. With a significance level of  = 0.05 in a left- tailed test,

15 Confidence Interval Estimate of p 1 – p 2 n1n1 n2n2 p 1 q 1 p 2 q 2 + ^ ^ ^ ^ where E = z   ( p 1 – p 2 ) – E < ( p 1 – p 2 ) < ( p 1 – p 2 ) + E ^^^^

16 Example: Use the same sample data to construct a 90% confidence interval estimate of the difference between the two population proportions. Note: 1─  = 0.90, so  = 0.10 and  = 0.05.

17 Example: Requirements are satisfied as we saw earlier. 90% confidence interval: z  /2 = Calculate the margin of error, E

18 Example: Construct the confidence interval

19 Final note: The confidence interval limits do not contain 0, implying that there is a significant difference between the two proportions. Thus the confidence interval, too, suggests that the fatality rate is lower for occupants in cars with air bags than for occupants in cars without air bags.

20 Press STAT and select TESTS Scroll down to 2-PropZTest press ENTER Type in x 1 : (number of successes in 1 st sample) n 1 : (number of trials in 1 st sample) x 2 : (number of successes in 2 nd sample) n 2 : (number of trials in 2 nd sample) choose H 1 : p 1 ≠p 2 p 2 (two tails) (left tail) (right tail) Press on Calculate Read test statistic z=… and P-value p=… Two proportions by TI-83/84

21 Press STAT and select TESTS Scroll down to 2-PropZInt press ENTER Type in x 1 : (number of successes in 1 st sample) n 1 : (number of trials in 1 st sample) x 2 : (number of successes in 2 nd sample) n 2 : (number of trials in 2 nd sample) C-Level: (confidence level) Press on Calculate Read the interval (…,…) Two proportions by TI-83/84

22 Section 9-3 Two Means: Independent Samples

23 Definitions Two samples are independent if the sample values selected from one population are not related to or somehow paired or matched with the sample values from the other population.

24   1 = population mean σ 1 = population standard deviation n 1 = size of the first sample = sample mean s 1 = sample standard deviation Corresponding notations for  2, σ 2, s 2, and n 2 apply to the second population. Notation for the first population:

25 Requirements 1. σ 1 an σ 2 are unknown and no assumption is made about the equality of σ 1 and σ The two samples are independent. 3. Both samples are random samples. 4. Either or both of these conditions are satisfied: The two sample sizes are both large (with n 1 > 30 and n 2 > 30) or both populations have normal distributions.

26 Tests for Two Means H 0 :  1 =  2 H 1 :  1   2, H 1 :  1  2 Note: no numerical values for  1 or  2 are claimed in the hypotheses. The goal is to compare the two means. two tails left tail right tail

27 Hypothesis Test for Two Means with Independent Samples: Test Statistic is Note:  1 –  2 =0 according to H 0 Degrees of freedom: df = smaller of n 1 – 1 and n 2 – 1.

28 Example: A headline in USA Today proclaimed that “Men, women are equal talkers.” That headline referred to a study of the numbers of words that men and women spoke in a day, see below. Use a 0.05 significance level to test the claim that men and women speak the same mean number of words in a day.

29 Example: Requirements are satisfied: two population standard deviations are not known and not assumed to be equal, independent samples, both samples are large. Step 1:Express claim as  1 =  2. Step 2:If original claim is false, then  1 ≠  2. Step 3:Alternative hypothesis does not contain equality, null hypothesis does. H 0 :  1 =  2 (original claim) H 1 :  1 ≠  2

30 Example: Step 4:Significance level is 0.05 Step 5:Use a t distribution Step 6:Calculate the test statistic

31 Example: Use Table A-3: area in two tails is 0.05, df = 185, which is not in the table, the closest value is t = ±1.972

32 Example: Step 7:Because the test statistic does not fall within the critical region, fail to reject the null hypothesis:  1 =  2 (or  1 –  2 = 0). Final conclusion: There is sufficient evidence to support the claim that men and women speak the same mean number of words in a day.

33 Confidence Interval Estimate of  1 –  2 : Independent Samples df = smaller n 1 – 1 and n 2 – 1 (x 1 – x 2 ) – E < (µ 1 – µ 2 ) < (x 1 – x 2 ) + E + n1n1 n2n2 s1s1 s2s2 where E = t   2 2

34 Example: Using the given sample data, construct a 95% confidence interval estimate of the difference between the mean number of words spoken by men and the mean number of words spoken by women.

35 Example: Find the margin of Error, E; use t  /2 = Construct the confidence interval use E = and

36 Press STAT and select TESTS Scroll down to 2-SampTTest press ENTER Select Input: Data or Stats. For Stats: Type in x 1 : (1 st sample mean) s x1 : (1 st sample st. deviation) n 1 : (1 st sample size) x 2 : (2 nd sample mean) s x2 : (2 nd sample st. deviation) n 2 : (2 nd sample size) choose H 1 :  1 ≠  2  2 (two tails) (left tail) (right tail) Tests about two means by TI-83/84

37 choose Pooled: No or Yes (always No) Tests about two means (continued) Press on Calculate Read the test statistic t=… and the P-value p=… Note: the calculator gives a more accurate P-value than the book does, because it uses a more accurate formula for degrees of freedom (see the line df=… in the calculator). The book adopts a simple but inaccurate rule df=smaller of n 1 -1 and n 2 -1.

38 Press STAT and select TESTS Scroll down to 2-SampTInt press ENTER Select Input: Data or Stats. For Stats: Type in x 1 : (1 st sample mean) s x1 : (1 st sample st. deviation) n 1 : (1 st sample size) x 2 : (2 nd sample mean) s x2 : (2 nd sample st. deviation) n 2 : (2 nd sample size) C-Level:  confidence level  Intervals for two means by TI-83/84

39 Press on Calculate Read the confidence interval (…,…) Note: the calculator gives a more accurate confidence interval than the book does, because it uses a more accurate formula for degrees of freedom (see the line df=… in the calculator). The book adopts a simple but inaccurate rule df=smaller of n 1 -1 and n choose Pooled: No or Yes (always No) Intervals for two means (continued)