Difference between revisions of "Hypothesis Testing"
(→A ProblemM. K. Chung's lecture notes, 2003.) |
(→A ProblemM. K. Chung's lecture notes, 2003.) |
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I believe that dogs are as smart as people. Assume IQ of a dog follows {{#tag:math|Xi \sim N(\mu,10^2)}}. IQ of 10 dogs are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100, 60. We want to test if dogs are as smart as people by testing | I believe that dogs are as smart as people. Assume IQ of a dog follows {{#tag:math|Xi \sim N(\mu,10^2)}}. IQ of 10 dogs are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100, 60. We want to test if dogs are as smart as people by testing | ||
− | <center>{{#tag:math|H_0 : \mu = 100 \text{vs.} H_1 : \mu < 100}}.</center> | + | <center>{{#tag:math|H_0 : \mu = 100 \text{ vs. } H_1 : \mu < 100}}.</center> |
One reasonable thing one may try is to see how high the sample mean is. | One reasonable thing one may try is to see how high the sample mean is. |
I believe that dogs are as smart as people. Assume IQ of a dog follows [math]Xi \sim N(\mu,10^2)[/math]. IQ of 10 dogs are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100, 60. We want to test if dogs are as smart as people by testing
One reasonable thing one may try is to see how high the sample mean is.
1 > x<-c(30, 25, 70, 110, 40, 80, 50, 60, 100, 60)
2 > mean(x)
3 [1] 62.5
Since the average IQ of 10 dogs are lower than 100, one would be inclined to reject [math]H_0[/math].
Let [math]\bar{X} [/math] be a test statistic and [math]R = (−∞,90][/math] to be a rejection region. Let’s compute the probability of making Type I error based on this testing procedure. Under the assumption [math]H_0[/math] is true,
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