One and two proportion z-intervals and tests, conditions, errors and power.
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- What is the null hypothesis in a one-proportion z-test?
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H_0: p = p_0, where p is the population proportion and p_0 is the hypothesized value.- What are the three possible alternative hypotheses for a one-proportion z-test?
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Less than (H_a: p < p_0), greater than (H_a: p > p_0), or two-sided (H_a: p not equal p_0).- What is the test statistic formula for a one-proportion z-test?
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z = (p_hat - p_0) / SE, where p_hat is the sample proportion and SE is the standard error.- What is the standard error formula for a one-proportion z-test?
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SE = sqrt[p_0(1 - p_0) / n], where p_0 is the hypothesized proportion and n is the sample size.- What is the formula for a one-proportion confidence interval?
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p_hat plus or minus z_star times sqrt[p_hat(1 - p_hat) / n], where p_hat is the sample proportion, z_star is the critical value, and n is the sample size.- What is the null hypothesis in a two-proportion z-test?
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H_0: p_1 = p_2, where p_1 and p_2 are the two population proportions.- What is the pooled proportion in a two-proportion z-test?
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p_c = (x_1 + x_2) / (n_1 + n_2), where x is the number of successes in each sample and n is the sample size.- What is the test statistic formula for a two-proportion z-test?
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z = (p_hat1 - p_hat2) / SE, where p_hat1 and p_hat2 are the sample proportions and SE is the standard error.- What is the standard error formula for the test statistic in a two-proportion z-test?
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SE = sqrt[p_c(1 - p_c)(1/n_1 + 1/n_2)], where p_c is the pooled proportion.- What is the standard error formula for a two-proportion confidence interval?
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SE = sqrt[p_hat1(1 - p_hat1)/n_1 + p_hat2(1 - p_hat2)/n_2], using individual sample proportions instead of pooled.- What is the formula for a two-proportion confidence interval?
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(p_hat1 - p_hat2) plus or minus z_star times SE, where SE uses the unpooled standard error formula.- What is the randomness condition for inference about proportions?
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Data must come from a random sample or randomized experiment to ensure observations are representative and independent.- What is the independence condition for one-proportion inference?
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The sample size must be no more than 10 percent of the population (n less than or equal 0.10N), ensuring independence between observations.- What is the sample size condition for one-proportion inference?
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Both the number of successes (np_0) and failures n(1 - p_0) must be at least 10.- What is the randomness condition for two-proportion inference?
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Both samples must come from random samples or randomized experiments.
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