First identify the proper confidence interval. Since we are dealing with one proportion, a 1-proportion Z-interval is appropriate. Now check to see if the necessary conditions are fulfilled. First is whether the data was collected from a simple random sample representative of the population. Second is whether n * p-hat and n * (1 - p-hat) are both sufficiently large, where n is the sample size and p-hat is the sample proportion. This is the case, since both 135 and 500 - 135 = 365 are both greater than 10, a generally accepted value. Third is whether n is a sufficiently small fraction of the population (about 1/10 of the population is the largest acceptable fraction). If you have at least 5000 students in your population, the test can be used. Finally, the calculation (assuming all conditions have been fulfilled). The confidence interval for a proportion is p-hat +/- z-star * sqrt(p-hat * (1 - p-hat) / n). Here z-star is the critical value for which P(abs(Z) > z-star) on the standard normal curve is equal to 1 minus your confidence level. In this case, we're looking for the value where the probability of a standard normal random variable producing a value either greater than z-star or less than negative z-star is 1 - .95 = 0.05. This value is approximately 1.96. Putting it all together: p-hat = 135/500 = 0.27
z-star = 1.96
n = 500
0.27 +/- 1.96 * sqrt(0.27 * 0.73 / 500)
0.27 +/- 1.96 * sqrt(0.0003942)
0.27 +/- 1.96 * .0199
0.27 +/- .03891
We are 95% confident that the true proportion of students who own computers is between .23109 and .30891.
There is a 95% probability that the true population proportion lies within the confidence interval.
What percentage of times will the mean (population proportion) not be found within the confidence interval?
Confidence intervals may be calculated for any statistics, but the most common statistics for which CI's are computed are mean, proportion and standard deviation. I have include a link, which contains a worked out example for the confidence interval of a mean.
confidence interval estimate
Why confidence interval is useful
The confidence interval becomes wider.
how are alpha and confidence interval related
No. The width of the confidence interval depends on the confidence level. The width of the confidence interval increases as the degree of confidence demanded from the statistical test increases.
The confidence interval is not directly related to the mean.
The confidence interval becomes smaller.
Estimated p = 75 / 250 = 0.3 Variance of proportion = p*(1-p)/n = 0.3(0.7)/250 =0.00084 S.D. of p is sqrt[0.00084] = 0.029 Confidence interval: phat-zval*sd = 0.3 - (1.96)(0.028983) phat-zval*sd = 0.3 + (1.96)(0.028983) Confidence interval is ( 0.2432 , 0.3568 )
No, it is not. A 99% confidence interval would be wider. Best regards, NS