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CHAPTER 9 TOOLS9.1

p = zẍ (see cumulative probabilities for sandard normal distribution ta

In the two tail test, the yielded p is multiplied by two.

Z Test Statistic for a Hypothesis Test of the Population Mean (when σ is known)ẍ (mean of sample) 5.7σ 1.8n 50μH0 (claimed mean) 5

zẍ 2.749859705

Critical Z Score (z a) 1.65 This is the extreme bound- find usi

p-value (from table) 0.0628 For two tail test, this is 1 minus the

Critical Sample Mean ( ẍa) 5.420021428

9.2Student T Distribution for the Mean ( σ unknown)TINV(alpha,degrees of freedom) finds critical values 2.093024μH0 210

α 0.05

n 20 xi

degrees of freedom 19 215 ẍ (Sample Mean) 205.8 129

s (sample standard dev) s=SQRT((Σ(xi - ẍ)2 )/(n-1) 84.208138 187

tα/2 (see table 5 in appendix) tẍ= (x-μH0)/(s/SQRT(n)) -0.223054 338p USE phStat 139ẍa 170.5898312 79

249.4101688 186305

55203163334

88

233261276225328201171

The p value is the probability of observing a sample mean at lehypothesis test, assuming the null hypothesis is true.

Sample

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9.3

Z Test for a Proportion When x and (n-x) are both at least equal to 5, the z p test statisti

π H0 0.37

x (data of interest) 31

n (sample size) 100p = x/n 0.31zp = (p - π H0) / SQRT((π H0*(1-π H0))/n) -1.242739532

Critical Sample Proportionszα/2 1.645

Pα = π H0 + (zα/2 )* √(π H0*(1-π H0))/n] 0.449421309 (for right side only)

Pα = π H0 - (zα/2 )* √(π H0*(1-π H0))/n] 0.290578691 (for left side only)

p-value = table value of z p (x2 if 2tail test)

9.4Probability of Type 2 Error Occuringn 36σ 5π H0 (hypothesized population mean) 58.4

α 0.01zα 2.33

ẍα = π H0 + (zα)*(σ/√n]) 60.34166667

μ (actual value) 61.6

z = (ẍα - μ)/(σ/√n]) -1.51β (probability of type 2 error) 0.5675 Equal to the standard normal distri

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Cumulative Probabilities for standard normal distribution tables are on page 858 and 859

ble)

g significance value α

zẍ table value multiplied by two. P=(1-zx)*2

α = significance level of the confidence interval degrees of freedom = n-1 n = sample

xi - ẍ (xi - ẍ)2 134729.29.2 84.64

-76.8 5898.24

-18.8 353.44

132.2 17476.84-66.8 4462.24 -0.18869

-126.8 16078.24

-19.8 392.0499.2 9840.64

-150.8 22740.64-2.8 7.84

-42.8 1831.84128.2 16435.24

-117.8 13876.84

27.2 739.8455.2 3047.0470.2 4928.0419.2 368.64

122.2 14932.84-4.8 23.04

-34.8 1211.04

ast as extreme as the one selected for the

tandard Dev. Table

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c follows a standard normal distribution.

ution table value of z

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size