8.6: Chapter 8 Formulas

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Hypothesis Test for One Mean

Use z-test when σ is given.

Use t-test when s is given.

If n < 30, population needs to be normal.

Type I Error-

Reject H₀ when H₀ is true.

Type II Error-

Fail to reject H₀ when H₀ is false.

Z-Test:

H₀: μ = μ₀

H₁: μ ≠ μ₀

\(Z=\frac{\bar{x}-\mu_{0}}{\left(\frac{\sigma}{\sqrt{n}}\right)}\)

TI-84: Z-Test

t-Test:

H₀: μ = μ₀

H₁: μ ≠ μ₀

\(t=\frac{\bar{x}-\mu_{0}}{\left(\frac{s}{\sqrt{n}}\right)}\)

TI-84: T-Test

z-Critical Values

Excel:

Two-tail: \(z_{\alpha / 2}\) = NORM.INV(1–\(\alpha\)/2,0,1)

Right-tail: \(z_{1-\alpha}\) = NORM.INV(1–\(\alpha\),0,1)

Left-tail: \(z_{\alpha}\) = NORM.INV(\(\alpha\),0,1)

TI-84:

Two-tail: \(z_{\alpha / 2}\) = invNorm(1–\(\alpha\)/2,0,1)

Right-tail: \(z_{1-\alpha}\) = invNorm(1–\(\alpha\),0,1)

Left-tail: \(z_{\alpha}\) = invNorm(\(\alpha\),0,1)

t-Critical Values

Excel:

Two-tail: \(t_{\alpha / 2}\) =T.INV(1–\(\alpha\)/2,df)

Right-tail: \(t_{1-\alpha}\) = T.INV(1–\(\alpha\),df)

Left-tail: \(t_{\alpha}\)= T.INV(\(\alpha\),df)

TI-84:

Two-tail: \(t_{\alpha / 2}\) = invT(1–\(\alpha\)/2,df)

Right-tail: \(t_{1-\alpha}\) = invT(1–\(\alpha\),df)

Left-tail: \(t_{\alpha}\) = invT(\(\alpha\),df)

Hypothesis Test for One Proportion

H₀: p = p₀

H₁: p ≠ p₀

\(Z=\frac{\hat{p}-p_{0}}{\sqrt{\left(\frac{p_{0} q_{0}}{n}\right)}}\)

TI-84: 1-PropZTest

Rejection Rules:

P-value method: reject H₀ when the p-value ≤ \(\alpha\).
Critical value method: reject H₀ when the test statistic is in the critical region (shaded tails).