Hypothesis Testing

Null Hypothesis H0 :Statementbeingtested;Claimaboutµ or historical value of µ Given Null Hypothesis: µ = k k is a value of the mean given µ is the population mean discussed throughout the worksheet Alternative Hypothesis H1 : Statement you will adopt in the situation in which evidence(data) is strong so H0 is rejected. Why do hypothesis testing? Sample mean may be di↵erent from the population mean.

Type of Test to Apply:

Right Tailed µ>kYou believe that µ is more than value stated in H0

Left-Tailed µ

Test µ When Known(P-Value Method) x¯ µ Given x is normal and is known: test statistic: z = x¯ /pn x¯ = mean of a random sample µ =valuestatedinH0 n =samplesize = population standard deviation ↵:Presetlevelofsigniﬁcance* *Note: ↵ is given in all of these approaches used

P-Values and Types of Tests:

Graph Test Conclusion 1. Left-tailed Test H : µ = kH: µ↵,wedonotrejectH . getting a test statistic as 0 zx¯ 0 low as or lower than zx¯ 2. Right-tailed Test H : µ = kH: µ>k 0 1 If P-value ↵,werejectH and say the data P-value = P (z>z) 0 x¯ are statistically signiﬁcant at the level ↵. This is the probability of If P-value >↵,wedonotrejectH . getting a test statistic as 0 0 zx¯ high as or higher than zx¯ 3. Two-tailed Test H : µ = kH: µ = k 0 1 6 P-value = 2P (z> zx¯ ) If P-value ↵,werejectH0 and say the data This is the probability| | of are statistically signiﬁcant at the level ↵. getting a test statistic ei- If P-value >↵,wedonotrejectH0. 0 ther lower than zx¯ or zx¯ zx¯ higher than z | | | | | | | x¯|

XXXX @csusm_stemcenter Tel: csusm.edu/stemsc STEM SC (N): (760) 750-4101 STEM SC (S): (760) 750-7324 Hypothesis Testing

Test µ When Unknown(P-Value Method Continued) x¯ µ test statistic: t = s=samplestandarddeviation s/pn

Critical Region Method x¯ µ Testing µ when is known: test statistic: z = = population standard deviation 0 /pn µ is the population mean discussed throughout the worksheet ↵:Presetlevelofsigniﬁcance Note: Assuming a table of areas to the left of z is used, area ↵ is converted to area 1-↵ if you are using the right tailed test. Also Note: Shaded area in ﬁgures below is the critical region.

Graph Method Conclusion Left-tailed Test If P-value ↵,werejectH . H : µ = kH: µ↵,wedonotrejectH . P-value = P (z critical value, fail to Find z-score of ↵: zc =-t reject H0. -t 0 Compare z0 to zc Right-tailed Test If P-value ↵,werejectH . H : µ = kH: µ>k 0 0 1 If P-value >↵,wedonotrejectH . P-value = P (z>t) 0 Given Area = ↵ If sample test statistic critical value, reject H Critical Value: 0 If sample test statistic< critical value, fail to Find z-score of ↵: zc =t reject H0. 0 t Compare z0 to zc Two-tailed Test If P-value ↵,werejectH . H : µ = kH: µ = k 0 0 1 If P-value >↵,wedonotrejectH . P-value = 2P (z> t6 ) 0 | | Area of each shaded region ↵ If sample test statistic lies at or beyond critical = 2 values, reject H0. Critical Value: If sample test statistic lies between critical val- Find z-score of ↵: zc =-t -t 0 t ues, fail to reject H0. (Notation: t

Note: For each formula to ﬁnd z-scores, if you can assume that x has a normal distribution, then any sample size n will work. If you cannot assume this, use a sample size n 30.

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