In Problems 7–12, test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test.
H0: p = 0.4versusH1: p ≠ 0.4n = 1000;x = 420;α = 0.01

Verified step by step guidance

1. Step 1: Verify the requirements for conducting a hypothesis test for a population proportion. Check that the sample size is large enough by ensuring both \( n \times p_0 \) and \( n \times (1 - p_0) \) are at least 10, where \( p_0 = 0.4 \). This confirms the sampling distribution of the sample proportion is approximately normal.
2. Step 2: Calculate the sample proportion \( \hat{p} \) using the formula \( \hat{p} = \frac{x}{n} \), where \( x = 420 \) and \( n = 1000 \).
3. Step 3: Compute the test statistic \( z \) using the formula for a population proportion test: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \] This measures how many standard errors the sample proportion is from the hypothesized proportion.
4. Step 4: (Classical approach) Determine the critical z-values for a two-tailed test at significance level \( \alpha = 0.01 \). These are the z-values that cut off the upper and lower \( \frac{\alpha}{2} \) tails of the standard normal distribution.
5. Step 5: (P-value approach) Calculate the P-value by finding the probability of observing a test statistic as extreme or more extreme than the calculated \( z \) under the null hypothesis. Since this is a two-tailed test, multiply the one-tail probability by 2. Then compare the P-value to \( \alpha \) to decide whether to reject \( H_0 \).

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis (H0) in favor of an alternative hypothesis (H1). It involves setting up H0 and H1, selecting a significance level (α), and using sample data to make a decision about the population parameter.

Classical Approach vs. P-value Approach

The classical approach compares a test statistic to critical values determined by the significance level to accept or reject H0. The P-value approach calculates the probability of observing the sample data under H0; if this probability is less than α, H0 is rejected. Both methods aim to assess evidence against H0.

Requirements for Testing a Population Proportion

Testing a population proportion requires that the sample size is large enough for the sampling distribution of the sample proportion to be approximately normal. This is typically verified by checking np and n(1-p) are both at least 5 or 10, ensuring the validity of the normal approximation used in the test.