11 One sample inference

# 11 One sample inference - Inference about means Because Y...

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Inference about means Z = Y ! μ " Y = Y ! n Because is normally distributed: Y But. .. We don’t know ! t = Y ! s / n A good approximation to the standard normal is then: t has a Student’s t distribution } Degrees of freedom df = n - 1

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Confidence interval for a mean Y ± SE Y t ! 2 ( ) , df SE Y = s n 95% confidence interval for a mean Example: Paradise flying snakes 0.9, 1.4, 1.2, 1.2, 1.3, 2.0, 1.4, 1.6 Undulation rates (in Hz) Estimate the mean and standard deviation Y = 1.375 s = 0.324 n = 8 Find the standard error Y ± SE Y t 2 ( ) , df SE Y = s n = 0.324 = 0.115
Find the critical value of t df = n ! 1 = 7 t 2 ( ) , df = t 0.05 2 ( ) ,7 = 2.36 Table C: Student's t distribution Putting it all together. .. Y ± SE Y t ! 2 ( ) , df = 1.375 ± 0.115 2.36 ( ) = 1.375 ± 0.271 1.10 < μ < 1.65 99% confidence interval t 2 ( ) , df = t 0.01 2 ( ) ,7 = 3.50 Y ± SE Y t 2 ( ) , df = 1.375 ± 0.115 3.50 ( ) = 1.375 ± 0.403 0.97 < μ < 1.78

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Confidence interval for the variance df s 2 ! " 2 , df 2 # \$ 2 # df s 2 1 % 2 , df 2 ! 2 Frequency ! 2 1- " /2 ! 2 " /2 2.5% 2.5% " = 0.05 95% confidence interval for the variance of flying snake undulation rate df s 2 2 , df 2 # 2 # df s 2 1 % 2 , df 2 df = n - 1 = 7 s 2 = (0.324) 2 = 0.105 2 , df 2 = 0.025,7 2 = 16.01 1 # 2 , df 2 = 0.975,7 2 = 1.69 df X 0.999 0.995 0.99 0.975 0.95 0.05 0.025 0.01 0.005 0.001 1 1.6 E-6 3.9E-5 0.00016 0.00098 0.00393
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11 One sample inference - Inference about means Because Y...

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