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

Verified step by step guidance

1. 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.
2. 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).
3. 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).
4. 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.
5. 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.

Key Concepts

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.