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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==A Problem<ref>M. K. Chung's lecture notes, 2003.</ref>== | ==A Problem<ref>M. K. Chung's lecture notes, 2003.</ref>== | ||
− | I believe that dogs are as smart as people. Assume IQ of a dog follows {{#tag:math| | + | I believe that dogs are as smart as people. Assume IQ of a dog follows {{#tag:math|X_i \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. | ||
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[1] 62.5 | [1] 62.5 | ||
</syntaxhighlight> | </syntaxhighlight> | ||
+ | |||
Since the average IQ of 10 dogs are lower than 100, one would be inclined to reject {{#tag:math|H_0}}. | Since the average IQ of 10 dogs are lower than 100, one would be inclined to reject {{#tag:math|H_0}}. | ||
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Let {{#tag:math|\bar{X} }} be a test statistic and {{#tag:math|R = (−∞,90]}} to be a rejection region. Let’s compute the probability of making Type I error based on this testing procedure. Under the assumption {{#tag:math|H_0}} is true, | Let {{#tag:math|\bar{X} }} be a test statistic and {{#tag:math|R = (−∞,90]}} to be a rejection region. Let’s compute the probability of making Type I error based on this testing procedure. Under the assumption {{#tag:math|H_0}} is true, | ||
+ | <center>{{#tag:math|X_i \sim N(100,10^2)}}</center> | ||
+ | |||
+ | Under this condition, {{#tag:math|\bar{X} \sim N(100, 10)}} and | ||
+ | |||
+ | <center>{{#tag:math|\alpha = P(\bar{X} \leq 90) }}</center> | ||
+ | |||
+ | <syntaxhighlight lang="r" line start="4"> | ||
+ | > pnorm(90,100,sqrt(10)) | ||
+ | [1] 0.0007827011 | ||
+ | </syntaxhighlight> | ||
I believe that dogs are as smart as people. Assume IQ of a dog follows [math]X_i \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,
Under this condition, [math]\bar{X} \sim N(100, 10)[/math] and
4 > pnorm(90,100,sqrt(10))
5 [1] 0.0007827011
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