2-sample t-test
ST 361 Ch8 Testing Statistical Hypotheses:
Testing Hypotheses about Means (§8.2-2) : Two-Sample t Test
Topics: Hypothesis testing with population means
► One-sample problem: Testing for a Population mean [pic]
1. Assume population SD is known: use a z test
2. Assume population SD is not known: use a t test
► Two-sample problem: : Testing for 2 population means [pic]
► A Special Case: the Paired t test
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► TWO-sample problem: Testing for 2 population means [pic]
o Motivating Example
Is there a difference between the life of batteries made by Duracell and Eveready? Let [pic] be the mean lifetime (in days) for Duracell batteries, and [pic] be that of Eveready batteries. Perform a 5% level of test.
| |Duracell |Eveready |
|n (batteries) |8 |10 |
|[pic] |41 |45 |
|Sample SDs |18 |20 |
Step 1: Specify the hypotheses
parameters of interest =[pic] and [pic]
[pic]
Step 2: significance level [pic]=0.05
Step 3: test statistic ????
Step 4: p-value
Step 5: conclusion: If p-value < [pic], then we reject [pic]and draw conclusion according to [pic]
Otherwise do not reject [pic], and draw conclusion according to [pic]
o Calculating the Test Statistics for Testing Two Means
▪ Need: Data are Normal.
▪ We will focus on the case that [pic] and [pic] are unknown (and may not be the same)
▪ Test statistic is
[pic]
Then this test statistic will have a t-distribution with the following df:
df = [pic] (round down!!!) [pic]
← Note that the test statistic should be consistent with your hypotheses. That is, if your hypotheses are stated in terms of [pic], then the corresponding test statistic should be
[pic]
(Back to the battery example)
Step 3: test statistic [pic]-0.45
[pic] (So we cannot use the normal table)
Step 4: p-value = [pic]0.694
Step 5: conclusion: Since p-value > the significance level, we don’t reject [pic]
Summary of the testing procedure for two population means:
1) Hypotheses
[pic] vs. [pic] (lower-tail test)
[pic] (upper-tail test)
[pic] (two-sided test)
2) Significance level
3) Test statistic
[pic]
With df = [pic] [pic]
4) P-value = [pic] if [pic]
[pic] if [pic]
[pic] = [pic] if [pic]
a) Conclusion: Reject [pic] if p-value [pic], and draw conclusion according to [pic] Otherwise do not reject [pic], and draw conclusion according to [pic]
Ex2. Mary can take either a scenic route to work or a non-scenic route. She decides that use of non-scenic route can be justified only if it reduces true average travel time by more than 10 min.
a) If [pic]refers to the average travel time via scenic route and [pic] to the average travel time via non-scenic route, what hypotheses should be tested?
b) What should be the test statistic for testing your hypothesis?
(1) [pic] (2) [pic]
(3) [pic] (4) [pic]
Ex3. Many people take ginkgo supplements advertised to improve memory. Are these over-the-counter supplements effective? In a study, elderly adults were assigned to the treatment group or control group. The 104 participants who were assigned to the treatment group took 40 mg of ginkgo 3 times a day for 6 weeks. The 115 participants assigned to the control group took a placebo pill 3 times a day for 6 weeks. At the end of 6 weeks, the Wechsler Memory Scale was administered. Higher scores indicate better memory function. Summary values are given in the following table:
| |N |[pic] |s |
|Ginkgo |104 |5.7 |0.6 |
|Placebo |115 |5.5 |0.5 |
Based on these results, is there evidence that taking 40mg of ginkgo 3 times a day is effective in increasing mean performance on the Wechsler Memory Scale? [pic]
Step 1: parameters of interest = [pic], the average memory score using Ginkgo, and [pic], the average memory score using placebo.
[pic]
Step 2: significance level is usually taken to be [pic]=0.05
Step 3: test statistic =
[pic]
Step 4: p-value = [pic]
Step 5: Conclusion: Since the p-value < significance level, we reject [pic] and conclude that Ginkgo does improve the memory score.
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