Setting Up Tests for Population Proportion - AP Statistics

What does the symbol $\alpha$ represent in hypothesis testing?

Level of significance. The probability threshold for rejecting the null hypothesis.
What is the meaning of a $p$ -value?

Probability of observing the data if $H_0$ is true. Measures how extreme the observed data is under the null assumption.
What symbol represents the sample proportion?

$\hat{p}$ . The observed proportion calculated from sample data.
What is the standard error formula for a sample proportion?

$\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ . Measures variability of the sample proportion estimate.
Which distribution is used for a population proportion hypothesis test?

Standard normal distribution. Uses $z$ -distribution when normal conditions are met.
State the null hypothesis for a test if the claim is $p > 0.3$ .

$H_0: p = 0.3$ . Null always states equality, regardless of the alternative claim direction.
Find the test statistic for $\hat{p} = 0.6$ , $p_0 = 0.5$ , $n = 100$ .

$z = 2.0$ . $z = \frac{0.6-0.5}{\sqrt{\frac{0.5(0.5)}{100}}} = \frac{0.1}{0.05} = 2.0$
What is the alternative hypothesis for a left-tailed test of population proportion?

$H_a: p < p_0$ . Tests if the proportion is less than the null value.
What is the purpose of setting up a test for a population proportion?

To determine if there is statistical evidence against $H_0$ . Evaluates whether sample data supports or contradicts a population claim.
What is the sample statistic in a proportion hypothesis test?

Sample proportion $\hat{p}$ . The proportion calculated from the sample data collected.
What is the formula for the test statistic in a population proportion test?

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$ . Standardizes the sample proportion using the null hypothesis assumption.
What is the formula for the confidence interval of a population proportion?

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ . Provides a range of plausible values for the population proportion.
Calculate the $p$ -value from a $z$ -score of 2.33 in a right-tailed test.

$p$ -value ≈ 0.0099. Right-tail probability from $z = 2.33$ using standard normal table.
What is the formula to calculate $z$ -score in hypothesis testing?

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

The hypothesized population proportion. The claimed value of the population proportion being tested.
Calculate the sample proportion if 45 out of 100 show a characteristic.

$\hat{p} = 0.45$ . Sample proportion equals number with characteristic divided by total.
What is the alternative hypothesis for a two-sided test of population proportion?

$H_a: p \neq p_0$ . Tests if the proportion differs from the null value in either direction.
Determine if $np_0 = 5$ and $n(1-p_0) = 5$ meet normal conditions.

No, conditions not satisfied. Both values must be at least 10 for normal approximation validity.
What is the alternative hypothesis for a right-tailed test of population proportion?

$H_a: p > p_0$ . Tests if the proportion is greater than the null value.
Identify the level of significance commonly used in hypothesis testing.

$\alpha = 0.05$ . Common threshold for Type I error probability.
State the alternative hypothesis for a claim $p < 0.4$ .

$H_a: p < 0.4$ . Alternative hypothesis matches the direction of the claim being tested.
What symbol represents the hypothesized population proportion?

$p_0$ . The specific value claimed for the population proportion in $H_0$ .
What is the population parameter of interest in a proportion test?

Population proportion $p$ . The true proportion in the entire population being studied.
What decision should be made if $p$ -value > $\alpha$ ?

Fail to reject the null hypothesis. Insufficient evidence to conclude the null hypothesis is false.
What is the decision rule for a hypothesis test?

Reject $H_0$ if $p$ -value < $\alpha$ . Compare probability to significance level to make conclusion.
What is the null hypothesis for a test of population proportion?

$H_0: p = p_0$ . States the hypothesized value for the population proportion parameter.
What is the condition for a normal approximation to be valid in a proportion test?

$np_0 \geq 10$ and $n(1-p_0) \geq 10$ . Ensures at least 10 successes and 10 failures for normal approximation.
What is Type II error in hypothesis testing?

Failing to reject $H_0$ when it is false. Missing a real effect that actually exists in the population.
What is the critical value for a 90% confidence level in a two-tailed test?

$z^* = 1.645$ . Bounds the middle 90% of the standard normal distribution.
Choose the correct test if $H_a: p \neq 0.25$ .

Two-tailed test. Not-equal alternative requires testing both tails of the distribution.
Find $z$ -score for $\hat{p} = 0.55$ , $p_0 = 0.5$ , $n = 200$ .

$z = 1.414$ . $z = \frac{0.55-0.5}{\sqrt{\frac{0.5(0.5)}{200}}} = \frac{0.05}{\sqrt{0.00125}} = 1.414$
Find the $z$ -score if $\hat{p} = 0.62$ , $p_0 = 0.5$ , $n = 150$ .

$z = 2.449$ . $z = \frac{0.62-0.5}{\sqrt{\frac{0.5(0.5)}{150}}} = \frac{0.12}{0.0408} = 2.449$
Provide an example of a two-tailed alternative hypothesis.

$H_a: p \neq 0.5$ . Tests whether proportion differs from 0.5 in either direction.
What is the critical value for a 95% confidence level in a two-tailed test?

$z^* = 1.96$ . Bounds the middle 95% of the standard normal distribution.
What does a $p$ -value less than $\alpha$ indicate?

Reject the null hypothesis. Strong evidence against $H_0$ when probability is below threshold.
Find and correct the error: Null hypothesis is $H_0: \mu = p_0$ .

Correct: $H_0: p = p_0$ . Null hypothesis should use $p$ for proportion, not $\mu$ for mean.
What is Type I error in hypothesis testing?

Rejecting $H_0$ when it is true. Incorrectly concluding there is an effect when none exists.
Calculate standard error for $\hat{p} = 0.7$ , $n = 100$ .

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

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