In a random sample of 684 observations,
p-bar = 0.3. What lower bound of a 90% confidence interval for p?
Round to four decimal places for all your calculations, including the FINAL answer.
The formula for a confidence interval for a proportion is:
p ± z*√(p*(1-p)/n)
where p is the sample proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size.
For a 90% confidence interval, the z-score is 1.645.
Plugging in the values, we get:
0.3 ± 1.645*√(0.3*(1-0.3)/684)
Simplifying:
0.3 ± 0.0493
The lower bound of the confidence interval is:
0.3 - 0.0493 = 0.2507
Rounded to four decimal places:
0.2507
To find the lower bound of a 90% confidence interval for the population proportion (p), we can use the formula:
Lower bound = sample proportion - margin of error
Given that the sample proportion (p-bar) is 0.3, we need to calculate the margin of error.
To calculate the margin of error, we use the formula:
Margin of error = critical value * standard error
The critical value is the number of standard deviations corresponding to the desired level of confidence, which is 90% in this case. For a 90% confidence level, the critical value is approximately 1.645.
The standard error is the estimated standard deviation of the sample proportion and is calculated using the formula:
Standard error = sqrt((p-bar * (1 - p-bar)) / n)
Where:
- p-bar is the sample proportion (0.3)
- n is the sample size (684)
Let's calculate the margin of error and the lower bound:
Standard error = sqrt((0.3 * (1 - 0.3)) / 684) ≈ 0.0170
Margin of error = 1.645 * 0.0170 ≈ 0.0279
Lower bound = 0.3 - 0.0279 ≈ 0.2721
Therefore, the lower bound of the 90% confidence interval for p is approximately 0.2721.