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STA101_lecture_11a

# STA101_lecture_11a - Testing hypotheses about the...

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Testing hypotheses about the population mean Consider the following problem (based on Keller 2006, exercise 9.22 p. 290): You know from past experience that the weight of a can of salmon is a normal random variable with a standard deviation of .18 ounce. The manufacturer claims that the net weight of a can has a mean of 6.05 ounces (or more): μ 6.05 We focus on the is equal to - part of the claim: μ = 6.05 A claim like this is called the null hypothesis (H 0 ) H 0 : μ = 6.05 What is the competing claim? μ …….. The competing claim is called the alternative hypothesis: H a : μ < 6.05 one -sided alternative: values of X <<< 6.05 are consistent with H a You draw a random sample of 36 cans and weigh them. The sample mean ( X ) is 5.97 ounces. Is this evidence consistent with H 0 ? Let us first draw the distribution that X would have if μ = 6.05 If the null hypothesis ( μ = 6.05) is true, X is normally distributed with: o a mean of μ x = μ = 6.05 ounces, and o a standard deviation of σ x = σ n = .18 36 = .03 ounce o normal_distribution_area.xls o hence, the probability that the sample mean is 5.97 ounces or less is: [Use normal_distribution_area.xls or TI-83: normcdf(low,up,mean,standard deviation) P( X 5.97 | μ = 6.05) = .0038 = .38% Comment on the claim made by the manufacturer. Answer: Given the low probability of finding a sample mean of 5.97 ounces or less, the claim appears to be false. we would tend to reject the claim; Is it possible that we rejected the null hypothesis while it is true ? Yes! Unlikely ( P = .38%), but possible. In that case, rejecting the null hypothesis would be a wrong decision. This decision error (rejecting the null hypothesis when it is true) is called a type I decision error . 1

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We just computed the probability of making a type I decision error : P (reject H 0 | H 0 is true) = P (type I error) = p -value = .38% Decision rule: Reject H 0 only if the probability of making a type I error is sufficiently small. Procedure: o Compute p -value = P (type I error) = P (reject H 0 | H 0 true) (example: p -value = .0038 = .38%) o Determine which maximum probability of making a type I error you are willing to tolerate: = significance level α Conventionnally, α = 5%, but see (*) below! o Decision rule: Reject H 0 if ( p -value) < (significance level α 29 . Report as: "As the p -value (.38%) is less than the significance level (5%), I can safely reject the null hypothesis that the mean weight is 6.05 ounce, in favor of the alternative hypothesis that it is less than 6.05 ounce." 2
[More details:] Hypotheses A hypothesis is a claim. Example of criminal trial: o Null hypothesis ( H 0 ): defendant is innocent o Alternative hypothesis ( H 1 ): defendant is guilty Jury decides (convict/acquit) on basis of evidence presented at trial. Decision errors Jury decides: Defendant is: Acquit (don't reject H 0 ) Convict (reject H 0 ) Innocent ( H 0 true) Right decision Type I error Guilty ( H 0 false) Type II error Right decision P (Type I error) = P ( reject H 0 | H 0 true) = p -value P (Type II error) = P (don't reject H 0 | H 0 false) In some legal systems, the jury can convict the defendant only if the evidence indicates guilt beyond a reasonable doubt . Discuss.

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STA101_lecture_11a - Testing hypotheses about the...

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