Final answer:
To determine if a sample is large enough, we check if np' and nq' are greater than or equal to 10. Sample b is large enough, while sample a is not. To construct a 90% confidence interval, use the formula: p' ± z * sqrt((p'*(1-p'))/n), where z is the critical value. For both samples, the confidence interval is [0.5763, 0.8237].
Step-by-step explanation:
To determine if a sample is large enough to construct a confidence interval for the population proportion, we need to check if the sample size satisfies certain conditions. One of the conditions is that np' and nq' must be greater than or equal to 10. Here, n represents the sample size and p' represents the sample proportion (i.e., number of successes divided by the sample size).
a. For sample a, n = 25 and p' = 0.7. So, np' = 25 * 0.7 = 17.5 and nq' = 25 * (1-0.7) = 7.5. Both np' and nq' are less than 10, so the sample size is not large enough to construct a confidence interval.
b. For sample b, n = 50 and p' = 0.7. So, np' = 50 * 0.7 = 35 and nq' = 50 * (1-0.7) = 15. Both np' and nq' are greater than 10, so the sample size is large enough to construct a confidence interval.
To construct a 90% confidence interval for the population proportion, we use the formula: p' ± z * √((p'*(1-p'))/n), where z is the critical value corresponding to the desired confidence level. For a 90% confidence interval, z is approximately 1.645.
a. For sample a, the point estimate is p' = 0.7. The error bound is given by: z * (√(p'*(1-p'))/n) = 1.645 * √((0.7*(1-0.7))/50) ≈ 0.1237. The confidence interval is [p' - error bound, p' + error bound] = [0.7 - 0.1237, 0.7 + 0.1237] = [0.5763, 0.8237].
b. For sample b, the point estimate is p' = 0.7. The error bound is given by: z * √((p'*(1-p'))/n) = 1.645 * √((0.7*(1-0.7))/50) ≈ 0.1237. The confidence interval is [p' - error bound, p' + error bound] = [0.7 - 0.1237, 0.7 + 0.1237] = [0.5763, 0.8237].