3 Lecture HYPOTHESIS TESTS FOR TWO MEANS
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CBA4 goes live in practice mode this week exam mode next week
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CBA4 goes live in practice mode this week exam mode next week Assignment 1 feedback
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CBA4 goes live in practice mode this week exam mode next week Assignment 1 feedback Mean 89.9 St. dev. 10.2 Median 94.0 IQR 11.1 95% CI (89.1, 90.9) Missing 27
Introduction
Last week you were introduced to the concept of hypothesis testing in statistics, and we considered hypothesis tests for the mean if we have a single sample drawn from a single population.
Introduction
Last week you were introduced to the concept of hypothesis testing in statistics, and we considered hypothesis tests for the mean if we have a single sample drawn from a single population. If we have two independent random samples from two populations, we can compare the two sample means in a test for two means (c.f. comparing one sample mean to a proposed value in the onesample case).
We use the same framework for hypothesis testing as for the onesample tests:
We use the same framework for hypothesis testing as for the onesample tests: 1. State the null hypothesis, H0 ;
We use the same framework for hypothesis testing as for the onesample tests: 1. State the null hypothesis, H0 ; 2. State the alternative hypothesis, H1 ;
We use the same framework for hypothesis testing as for the onesample tests: 1. State the null hypothesis, H0 ; 2. State the alternative hypothesis, H1 ; 3. Calculate a test statistic;
We use the same framework for hypothesis testing as for the onesample tests: 1. State the null hypothesis, H0 ; 2. State the alternative hypothesis, H1 ; 3. Calculate a test statistic; 4. Find the pvalue, and
We use the same framework for hypothesis testing as for the onesample tests: 1. State the null hypothesis, H0 ; 2. State the alternative hypothesis, H1 ; 3. Calculate a test statistic; 4. Find the pvalue, and 5. Use table 2.1 to form your conclusions.
We use the same framework for hypothesis testing as for the onesample tests: 1. State the null hypothesis, H0 ; 2. State the alternative hypothesis, H1 ; 3. Calculate a test statistic; 4. Find the pvalue, and 5. Use table 2.1 to form your conclusions. However, the calculations required for the test statistic in step 3 are slightly different.
Testing two means
Recall that, in the test for one mean, there were two cases: population variance ( 2 ) known and population variance unknown.
Testing two means
Recall that, in the test for one mean, there were two cases: population variance ( 2 ) known and population variance unknown. Similarly, when comparing two means, we can consider case 1: both population variances known and
Testing two means
Recall that, in the test for one mean, there were two cases: population variance ( 2 ) known and population variance unknown. Similarly, when comparing two means, we can consider case 1: both population variances known and case 2: both population variances unknown.
2 2 Both population variances (1 and 2 ) known
1. State the null hypothesis This time, the null hypothesis is H0 : 1 = 2 , i.e. the two population means are equal.
2 2 Both population variances (1 and 2 ) known
1. State the null hypothesis This time, the null hypothesis is H0 : 1 = 2 , i.e. the two population means are equal. 2. State the alternative hypothesis We usually test against the (twotailed) alternative: H1 : 1 = 2 , i.e. the population means are not equal. However, we might use the onetailed alternatives:
2 2 Both population variances (1 and 2 ) known
1. State the null hypothesis This time, the null hypothesis is H0 : 1 = 2 , i.e. the two population means are equal. 2. State the alternative hypothesis We usually test against the (twotailed) alternative: H1 : 1 = 2 , i.e. the population means are not equal. However, we might use the onetailed alternatives: H1 : 1 > 2 , or
2 2 Both population variances (1 and 2 ) known
1. State the null hypothesis This time, the null hypothesis is H0 : 1 = 2 , i.e. the two population means are equal. 2. State the alternative hypothesis We usually test against the (twotailed) alternative: H1 : 1 = 2 , i.e. the population means are not equal. However, we might use the onetailed alternatives: H1 : 1 > 2 , H1 : 1 < 2 . or
3. Calculate the test statistic The test statistic for a twosample test (when both population variances are known) is z = |1 - x2 | x
2 1 n1
+
2 2 n2
3. Calculate the test statistic The test statistic for a twosample test (when both population variances are known) is z = |1 - x2 | x
2 1 n1
+
2 2 n2
4. Find the pvalue This is found from statistical tables; since, in this case, both 2 2 population variances 1 and 2 are known, we refer to standard normal tables. As before, we find a range for our pvalue by comparing our test statistic to the 10%, 5% and 1% critical values.
5. Form a conclusion Exactly the same again! Use table 2.1 to help you decide what to do! Word your conclusions in the context of the original question.
Example (page 25)
Before a training session for call centre employees a sample of 50 calls to the call centre had an average duration of 5 minutes, whereas after the training session a sample of 45 calls had an average duration of 4.5 minutes. The population variance is known to have been 1.5 minutes before the course and 2 minutes afterwards. Has the course been effective?
Example (page 25)
Steps 1 and 2 (hypotheses) We test
Example (page 25)
Steps 1 and 2 (hypotheses) We test H0 : B = A against
Example (page 25)
Steps 1 and 2 (hypotheses) We test H0 : B = A H1 : B = A against
Example (page 25)
Step 3 (test statistic) Since both population variances are known, we use
Example (page 25)
Step 3 (test statistic) Since both population variances are known, we use z = |B - xA | x
2 B nB
+
2 A nA
Example (page 25)
Step 3 (test statistic) Since both population variances are known, we use z = |B - xA | x
2 B nB
+
2 A nA
=
|5 - 4.5|
1.5 50
+
2 45
Example (page 25)
Step 3 (test statistic) Since both population variances are known, we use z = |B - xA | x
2 B nB
+
2 A nA
=
|5 - 4.5|
1.5 50
+
2 45
= 1.833
Example (page 25)
Step 4 (pvalue) We used a twotailed alternative; using table 2.2, we get:
Example (page 25)
Step 4 (pvalue) We used a twotailed alternative; using table 2.2, we get: Significance level Critical value 10% 1.645 5% 1.96 1% 2.576
Example (page 25)
Step 4 (pvalue) We used a twotailed alternative; using table 2.2, we get: Significance level Critical 10% value 1.645 5% 1.96 1% 2.576
Since z = 1.833 lies between 1.645 and 1.96, our pvalue lies between 5% and 10%.
Example (page 25)
Step 5 (conclusion) Using table 2.1 (last week), this means that:
Example (page 25)
Step 5 (conclusion) Using table 2.1 (last week), this means that: we have slight evidence against H0
Example (page 25)
Step 5 (conclusion) Using table 2.1 (last week), this means that: we have slight evidence against H0 This is not small enough to reject H0 , and so we retain H0
Example (page 25)
Step 5 (conclusion) Using table 2.1 (last week), this means that: we have slight evidence against H0 This is not small enough to reject H0 , and so we retain H0 There is insufficient evidence to suggest that the training has had any affect on the average duration of a call.
2 2 Both population variances (1 and 2 ) unknown
In the more likely situation where the population variances are unknown, the test statistic becomes
2 2 Both population variances (1 and 2 ) unknown
In the more likely situation where the population variances are unknown, the test statistic becomes t = |1 - x2 | x s
1 n1
+
1 n2
,
2 2 Both population variances (1 and 2 ) unknown
In the more likely situation where the population variances are unknown, the test statistic becomes t = |1 - x2 | x s
1 n1
+
1 n2
,
where s is a "pooled standard deviation", and is found as
2 2 Both population variances (1 and 2 ) unknown
In the more likely situation where the population variances are unknown, the test statistic becomes t = |1 - x2 | x s
1 n1
+
1 n2
,
where s is a "pooled standard deviation", and is found as s =
2 2 (n1 - 1)s1 + (n2 - 1)s2 . n1 + n2 - 2
2 2 Both population variances (1 and 2 ) unknown
In the more likely situation where the population variances are unknown, the test statistic becomes t = |1 - x2 | x s
1 n1
+
1 n2
,
where s is a "pooled standard deviation", and is found as s =
2 2 (n1 - 1)s1 + (n2 - 1)s2 . n1 + n2 - 2
Like before, we have to use ttables to obtain our critical value; the degrees of freedom is now found as = n1 + n2 - 2.
Example (page 26)
A company is interested in knowing if two branches have the same level of average transactions. The company sample a small number of transactions and calculates the following statistics: Shop 1 Shop 2 x1 = 130 x2 = 120
2 s1 = 700 2 s2 = 800
n1 = 12 n2 = 15
Test whether or not the two branches have (on average) the same level of transactions.
Example (page 26)
Steps 1 and 2 (hypotheses) Our null and alternative hypotheses are:
Example (page 26)
Steps 1 and 2 (hypotheses) Our null and alternative hypotheses are: H0 : 1 = 2 versus
Example (page 26)
Steps 1 and 2 (hypotheses) Our null and alternative hypotheses are: H0 : 1 = 2 H1 : 1 = 2 . versus
Example (page 26)
Step 3 (calculating the test statistic) Since both population variances are unknown (only the sample values are given), the test statistic is t = |1 - x2 | x s
1 n1
+
1 n2
;
Example (page 26)
Step 3 (calculating the test statistic) Since both population variances are unknown (only the sample values are given), the test statistic is t = |1 - x2 | x s
1 n1
+
1 n2
;
thus, we first need to obtain the pooled variance s. This is given as
Example (page 26)
Step 3 (calculating the test statistic) Since both population variances are unknown (only the sample values are given), the test statistic is t = |1 - x2 | x s
1 n1
+
1 n2
;
thus, we first need to obtain the pooled variance s. This is given as s =
2 2 (n1 - 1)s1 + (n2 - 1)s2 n1 + n2 - 2
Example (page 26)
Step 3 (calculating the test statistic) Since both population variances are unknown (only the sample values are given), the test statistic is t = |1 - x2 | x s
1 n1
+
1 n2
;
thus, we first need to obtain the pooled variance s. This is given as s = =
2 2 (n1 - 1)s1 + (n2 - 1)s2 n1 + n2 - 2
11 700 + 14 800 25
Example (page 26)
St...