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chap105

# Thecalculationisillustratedinthegraphbelow pdfof x

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Unformatted text preview: pulation standard deviation is: σ = 3 . The decision rule adopted is to reject the null hypothesis in favour of the alternative if the calculated sample mean is such that: x < 30.8 (a) With a sample size of n = 36, what is the probability of a Type I error, using this decision rule ? α = P( X < 30 .8 μ = 32 ) This is the probability that the sample mean is in the rejection region when the null hypothesis is true (the true mean is 32). This is found as: α ⎛ X − 32 30 .8 − 32 ⎞ ⎟ = P⎜ ⎜ σ n < 3 36 ⎟ ⎠ ⎝ − 1 .2 ⎞ = P⎛ Z < ⎜ ⎟ 3/6 ⎠ ⎝ = P( Z < − 2 . 4 ) = 1 − P( Z < 2 . 4 ) = 1 − 0.9918 look ‐ up in Appendix Table 1 = 0.0082 7 Chapter 10.5 (b) With a sample size of n = 9, what is the probability of a Type I error, using this decision rule ? The calculation is illustrated in the graph below. PDF of X with μ = 32 , σ = 3 n = 36 n=9 30.8 32 Reject H0 ← → d o n o t r e j e c t With n = 9 the probability that the sample mean is in the rejection region (the lower tail) is greater than for the same test based on n = 36. That is, the probability of a Type I error is greater for a smaller sample size. A numerical answer...
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