In Exercises 45–48, determine whether a normal sampling distribution can be used to approximate the binomial distribution. If it can, test the claim.

Claim: p≥0.04; α=0.10
Sample statistics: p_hat = 0.03, n=30

검증된 단계별 안내

Step 1: Verify if the normal approximation to the binomial distribution can be used. For this, check the conditions: (1) np ≥ 5 and (2) n(1-p) ≥ 5, where p is the hypothesized population proportion (p = 0.04) and n is the sample size (n = 30). Calculate np and n(1-p) to confirm.

Step 2: If the conditions for normal approximation are satisfied, define the null hypothesis (H₀) and the alternative hypothesis (H₁). For this problem, H₀: p ≥ 0.04 and H₁: p < 0.04 (since the claim is p ≥ 0.04, the test is one-tailed).

Step 3: Calculate the test statistic using the formula for a z-test for proportions: z = (p̂ - p) / √(p(1-p)/n), where p̂ is the sample proportion (p̂ = 0.03), p is the hypothesized proportion (p = 0.04), and n is the sample size (n = 30).

Step 4: Determine the critical value for the z-test at the given significance level (α = 0.10) for a one-tailed test. Use a z-table or statistical software to find the critical z-value corresponding to α = 0.10.

Step 5: Compare the calculated z-test statistic to the critical z-value. If the test statistic is less than the critical value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.

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주요 개념

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Normal Approximation to the Binomial Distribution

The normal approximation to the binomial distribution is applicable when certain conditions are met, specifically when both np and n(1-p) are greater than or equal to 5. This allows the binomial distribution, which is discrete, to be approximated by a normal distribution, which is continuous, facilitating easier calculations for probabilities and hypothesis testing.

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 about the population proportion is being tested.

Sample Proportion (p_hat)

The sample proportion (p_hat) is the ratio of the number of successes in a sample to the total number of observations in that sample. It serves as an estimate of the population proportion (p). In this scenario, p_hat = 0.03 indicates that 3% of the sample exhibited the characteristic of interest, which is crucial for evaluating the claim regarding the population proportion.

In Exercises 45–48, determine whether a normal sampling distribution can be used to approximate the binomial distribution. If it can, test the claim.Claim: p≥0.04; α=0.10Sample statistics: p_hat = 0.03, n=30