STA 100 Midterm 2 Flashcards

5 Steps to Hypothesis Testing

1) State the Null and Alternative Hypothesis

2) Calculate the Test Statistic

3) Calculate the P-Value

4) State Decision Rules

• Compare test-statistic to critical value

• Compare P-value to a

• Compared hypothesized value to confidence interval

5) State your conclusion and interpret in the context of the problem

Null Hypothesis (H₀)

The statement which we assume to be true. Always has the equality sign (=, ≤, ≥)

Alternative Hypothesis (H_A)

The claim that we wish to test or show. Always has the inequality sign (≠, <, > )

Possible Hypothesis Tests

Two-sided:

• H₀: μ = μ₀

• H_A: μ ≠ μ₀

One-sided:

• H₀: μ ≤ μ₀

• H_A: μ > μ₀

One-sided:

• H₀: μ ≥ μ₀

  • H_A: μ < μ₀

Test-Statistic

Measures how much our sample data differs from H₀

We assume a value for μ₀ based on a large prior study or reasonable logic

(Tests how far the sample mean is from the hypothesized mean)

• The larger our test statistic is, the more our sample differs from H₀

P-value

Probability, assuming Null Hypothesis (H₀) were true, of observing our test-statistic or more extreme calculated from data, the direction depends on the Alternative Hypothesis (H_A)

Reject H₀ if

Test statistic is greater than critical value

P-value is less than a

μ₀ not in confidence interval

Errors in Hypothesis Testing

Type I Error: Reject the null when in reality the null is true (probability = a (alpha))

Type II Error: Fail to reject the null when in reality the null is false (probability = B (beta))

We can control error, but we reduce one error the other just grows bigger

Usually we control Type I error

Confidence Interval for (μ₁ - μ₂)

If both bounds are below zero, it gives evidence that μ₁ < μ₂ and the distributions do not overlap (much if at all)

If both bounds are above zero, it gives evidence that μ₁ > μ₂ and the distributions do no overlap (much if at all)

If the bounds cover zero, μ₁ = μ₂ or μ₁ - μ₂ = 0 is plausible value so there is no significant difference between μ₁ and μ₂ and the distributions overlap a lot

Paired t-test

When two observations occur in pairs (not independent)

Analyze the differences (D_i = Y_i1 - Y_i2)

Mean of the differences is equal to the difference of the means

• μ_D = μ₁ - μ₂

ANOVA (Analysis of Variance)

Used to compare means of multiple groups to the overall mean, regardless of group, and to consider the variances of each group

Comparing Means of Many Independent Groups

Case 1:

• Large Difference in Means

• Small variances within

• Almost no overlap

• Significantly different

Case 2:

• Smaller Differences in Means

• Bigger variances within

• A lot of overlap

• Not significantly different

ANOVA Terms

SSTO = SS(total) = Sums of Squares Total

SSB = SS(between) = Sums of Squares of Between groups

SSW = SS(within) = Sums of Squares Within groups

SSTO = SSB + SSW

Between variance = MSB = SSB/(I-1)

Within variance = MSW = SSW/(n.-I) = s_p² = pooled variance

Total variance = MSTO = SSTO/(n.-1) = overall sample variance

MSTO ≠ MSB + MSW

ANOVA Analysis

If MSB/MSW is large → significant difference in group means

If MSB/MSW is small → no significant difference in group means

Global F-test for ANOVA

H₀ = μ_{1 =} μ_{2 =} … = μ₁

H_A = At least one μ_i differs

ANOVA Assumptions

• Random samples are taken from each I group

• The I samples must be independent of each other

• The I populations are Normally distributed

• The I populations have equal variances

ANOVA Confidence Intervals (Multiple Comparisons)

If we rejected the null, we should find which means differ

P(At least one Type I Error) = 1-(1-a)^k

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