Setting Up Tests for Population Proportion - AP Statistics

$\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Measures variability of the sample proportion estimate.

$np_0 \geq 10$ and $n(1-p_0) \geq 10$. Ensures at least 10 successes and 10 failures for normal approximation.

$p_0$. The specific value claimed for the population proportion in $H_0$.

$H_a: p < p_0$. Tests if the proportion is less than the null value.

$H_a: p \neq p_0$. Tests if the proportion differs from the null value in either direction.

$H_0: p = p_0$. States the hypothesized value for the population proportion parameter.

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$. Standardizes the sample proportion using the null hypothesis assumption.

$z^* = 1.96$. Bounds the middle 95% of the standard normal distribution.

$H_a: p < 0.4$. Alternative hypothesis matches the direction of the claim being tested.

Reject $H_0$ if $p$-value < $\alpha$. Compare probability to significance level to make conclusion.

$H_a: p \neq 0.5$. Tests whether proportion differs from 0.5 in either direction.

The hypothesized population proportion. The claimed value of the population proportion being tested.

$0.0458$. $SE = \sqrt{\frac{0.7(1-0.7)}{100}} = \sqrt{\frac{0.21}{100}} = 0.0458$

No, conditions not satisfied. Both values must be at least 10 for normal approximation validity.

$z = 1.414$. $z = \frac{0.55-0.5}{\sqrt{\frac{0.5(0.5)}{200}}} = \frac{0.05}{\sqrt{0.00125}} = 1.414$

$H_0: p = 0.3$. Null always states equality, regardless of the alternative claim direction.

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$. Standardizes the sample proportion using the null hypothesis assumption.

Level of significance. The probability threshold for rejecting the null hypothesis.

$H_a: p \neq 0.5$. Tests whether proportion differs from 0.5 in either direction.

Failing to reject $H_0$ when it is false. Missing a real effect that actually exists in the population.

$\hat{p} = 0.45$. Sample proportion equals number with characteristic divided by total.

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$. Standardizes the sample proportion for hypothesis testing.

The hypothesized population proportion. The claimed value of the population proportion being tested.

No, conditions not satisfied. Both values must be at least 10 for normal approximation validity.

$H_0: p = 0.3$. Null always states equality, regardless of the alternative claim direction.

$H_a: p < 0.4$. Alternative hypothesis matches the direction of the claim being tested.

Population proportion $p$. The true proportion in the entire population being studied.

$p$-value ≈ 0.0099. Right-tail probability from $z = 2.33$ using standard normal table.

Sample proportion $\hat{p}$. The proportion calculated from the sample data collected.