In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p ≠0.15, α=0.05. Sample statistics: p_hat = 0.12, n=500

검증된 단계별 안내

Step 1: Verify the conditions for using a normal sampling distribution. Specifically, check if the sample size is large enough by ensuring that both n * p and n * (1 - p) are greater than or equal to 10. Use the claimed population proportion p = 0.15 and the sample size n = 500.

Step 2: Calculate the standard error (SE) of the sample proportion using the formula: SE = sqrt((p * (1 - p)) / n). Substitute p = 0.15 and n = 500 into the formula.

Step 3: Compute the z-score to test the claim. Use the formula: z = (p_hat - p) / SE, where p_hat = 0.12 is the sample proportion, p = 0.15 is the claimed population proportion, and SE is the standard error calculated in Step 2.

Step 4: Determine the critical z-values for a two-tailed test at the significance level α = 0.05. These critical values correspond to the points where the cumulative probability is 0.025 in each tail of the standard normal distribution.

Step 5: Compare the calculated z-score from Step 3 to the critical z-values from Step 4. If the z-score falls outside the range of the critical values, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Conclude whether there is sufficient evidence to support the claim that p ≠ 0.15.

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주요 개념

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Normal Sampling Distribution

A normal sampling distribution is a probability distribution of sample means or proportions that approaches a normal distribution as the sample size increases, typically due to the Central Limit Theorem. For proportions, the distribution can be considered normal if both np and n(1-p) are greater than 5, ensuring that the sample size is sufficiently large to approximate normality.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1, based on a predetermined significance level (α).

Significance Level (α)

The significance level (α) is the threshold used in hypothesis testing to determine whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common values for α are 0.05 and 0.01, indicating a 5% or 1% risk of concluding that a difference exists when there is none.

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.Claim: p ≠0.15, α=0.05. Sample statistics: p_hat = 0.12, n=500