In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p <0.12, α=0.01. Sample statistics: p_hat = 0.10, n=40

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 5. Use the claimed population proportion p = 0.12 for this calculation.

Step 2: Calculate the standard error (SE) of the sample proportion using the formula: SE = sqrt((p * (1 - p)) / n), where p = 0.12 and n = 40.

Step 3: Compute the z-test statistic using the formula: z = (p_hat - p) / SE, where p_hat = 0.10, p = 0.12, and SE is the standard error calculated in Step 2.

Step 4: Determine the critical value for a one-tailed test at the significance level α = 0.01. Use a z-table or standard normal distribution to find the critical z-value corresponding to α = 0.01.

Step 5: Compare the calculated z-test statistic from Step 3 to the critical z-value from Step 4. If the z-test statistic is less than the critical z-value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim p < 0.12.

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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 approximated as normal if both np and n(1-p) are greater than 5, ensuring that the sample size is sufficiently large to represent the population accurately.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population 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). In this case, the claim is that the population proportion p is less than 0.12.

Significance Level (α)

The significance level (α) is the threshold for determining whether to reject the null hypothesis in hypothesis testing. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that p < 0.12 when it is actually not true, thus requiring strong evidence from the sample data.

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.Claim: p <0.12, α=0.01. Sample statistics: p_hat = 0.10, n=40