Establish a 99% confidence interval for the population proportion if a sample of 200 had a sample proportion of 42%.
To establish a 99% confidence interval for the population proportion, we will use the formula:
Confidence Interval = (Sample Proportion) ± (Z-value) * √[(Sample Proportion * (1 - Sample Proportion)) / Sample Size]
First, let's calculate the Z-value. For a 99% confidence level, the Z-value is found from the Z-table or can be determined using a statistical calculator. The Z-value for a 99% confidence level is approximately 2.576.
Next, we will substitute the given values into the formula. The sample proportion is 42% (or 0.42) and the sample size is 200.
Confidence Interval = 0.42 ± 2.576 * √[(0.42 * (1 - 0.42)) / 200]
Now we can calculate the lower and upper bounds of the confidence interval:
Lower Bound = 0.42 - (2.576 * √[(0.42 * (1 - 0.42)) / 200])
Upper Bound = 0.42 + (2.576 * √[(0.42 * (1 - 0.42)) / 200])
After performing the calculations, we obtain the following 99% confidence interval:
Lower Bound = 0.3609
Upper Bound = 0.4791
Therefore, we can say with 99% confidence that the population proportion lies within the interval of 0.3609 to 0.4791.