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| 4 assumptions of sampling distribution of the difference between two proportions |
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Definition
1. Sampling distribution of p1hat-p2hat is normal
2. And mean mew(p1hat-p2hat)=p1-p2
3. n1p1hat(1-p1hat)>=10
4. n2p2hat(1-p2hat)>=10 |
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| [(p1(1-p1)/n1)+(p2(1-p2)/n2)]^0.5 |
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| [(p1hat-p2hat)-(p1-p2)]/[(p1(1-p1)/n1)+(p2(1-p2)/n2)]^0.5 |
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| The best point estimate of p because p is unknown |
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| When assuming null is true, p1-p2 = 0 so the new z0 = |
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| (p1hat-p2hat)/[p(1-p)]^0.5 x [(1/n1)+(1/n2)]^0.5 |
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| Pooled estimate of p, phat = |
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| When we substitute pooled estimate of p into z0 equation, we get test statistic two independent population proportions, z0 = |
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| (p1hat-p2hat)/[phat(1-phat)]^0.5 x [(1/n1)+(1/n2)]^0.5 |
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| 3 assumptions for hypothesis test regarding the difference between two population proportions |
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Definition
1. The samples are indecently obtained using simple random sampling
2. n1p1hat(1-p1hat) > 10 and n2p2hat(1-p2hat) > 10
3. The sample size is no more than 5% of the population size |
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Two-tailed test, two population proportions
H0:
H1: |
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Definition
H0: p1 = p2
H1: p1 not=to p2 |
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Left-tailed test, two population proportions
H0:
H1: |
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Right-tailed test, two population proportions
H0:
H1: |
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Two-tailed test, classical approach
If z0 > z(alpha/2), then |
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Left-tailed test, classical approach
If z0 < -zalpha, then |
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Right-tailed test, classical approach
If z0 > zalpha, then |
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| 3 assumptions for constructing a confidence interval for difference between two population proportions |
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Definition
1. The samples are indecently obtained using simple random sampling
2. n1p1hat(1-p1hat) > 10 and n2p2hat(1-p2hat) > 10
3. The sample size is no more than 5% of the population size |
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| Lower bound CI, two population proportions = |
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Definition
| (p1hat-p2hat) - z(alpha/2) x [(p1hat(1-p1hat)/n1)+(p2hat(1-p2hat)/n2)]^0.5 |
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| Upper bound CI, two population proportions = |
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Definition
| (p1hat-p2hat) + z(alpha/2) x [(p1hat(1-p1hat)/n1)+(p2hat(1-p2hat)/n2)]^0.5 |
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| 2 assumptions for testing a hypothesis regarding the difference of two proportions with dependent samples (McNemar's Test) |
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Definition
1. The samples are dependent and are obtained randomly
2. The total number of observations where the outcomes differ must be greater than or equal to 10 (f12+f21>=10) |
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| Test statistic, two matched pairs proportions z0 = |
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| [abs val(f12-f21) - 1]/(f12+f21)^0.5 |
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| Margin of error for population proportions, E = |
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Definition
| z(alpha/2) x [(p1hat(1-p1hat)/n)+(p2hat(1-p2hat)/n2)]^0.5 |
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| Sample size for estimating p1-p2, n = n1 = n2 = |
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| [p1hat(1-p1hat) + p2hat(1-p2hat)](z(alpha/2)/E)^2 |
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| Sample size for estimating p1-p2 is p1 and p2 are unavailable, n = n1 = n2 = |
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