In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225

검증된 단계별 안내

Step 1: Verify the conditions for using a normal sampling distribution. The two conditions are: (1) The sample size n must be large enough such that both n * p and n * (1 - p) are greater than or equal to 5, where p is the hypothesized population proportion. (2) The sampling must be random and independent.

Step 2: Calculate the standard error (SE) of the sampling distribution of the sample proportion using the formula: SE = sqrt((p * (1 - p)) / n), where p is the hypothesized population proportion and n is the sample size.

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

Step 4: Determine the critical value for the given significance level α = 0.04. Since the claim is one-tailed (p > 0.70), find the z-critical value corresponding to the upper tail of the standard normal distribution.

Step 5: Compare the z-test statistic to the z-critical value. If the z-test statistic is greater than the z-critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.

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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, according to the Central Limit Theorem. For proportions, this is applicable when 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 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).

Significance Level (α)

The significance level (α) is the threshold for determining whether a result is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In this case, α=0.04 indicates a 4% risk of making such an error, guiding the decision-making process in hypothesis testing.

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.Claim: p > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225