An organization sends a random sample of surveys to 100,000 people and received 4,781 responses. A) Give a 95% confidence interval for the true proportion of those from their entire mailing list who may donate.
To calculate the 95% confidence interval for the true proportion of people who may donate, we can use the following formula:
Confidence interval = sample proportion ± (critical value) * (standard error)
Step 1: Calculate the sample proportion
The sample proportion is the number of people who responded divided by the total number of surveys sent. In this case, it is 4,781/100,000 = 0.04781.
Step 2: Calculate the critical value
The critical value is obtained from a standard normal distribution table or by using a statistical calculator. For a 95% confidence level, the critical value is approximately 1.96.
Step 3: Calculate the standard error
The standard error is the standard deviation of the sampling distribution, which can be calculated using the following formula:
Standard error = sqrt [(sample proportion * (1 - sample proportion)) / sample size]
In this case, the sample size is 100,000. Plugging in the values, the standard error can be calculated as follows:
Standard error = sqrt [(0.04781 * (1 - 0.04781)) / 100,000]
Step 4: Calculate the confidence interval
Now that we have all the values, we can calculate the confidence interval:
Confidence interval = 0.04781 ± (1.96 * standard error)
Keep in mind that the ± symbol represents "plus or minus." Let's plug in the values:
Confidence interval = 0.04781 ± (1.96 * standard error)
Confidence interval = 0.04781 ± (1.96 * 0.00215)
Finally, calculate the upper and lower bounds of the confidence interval:
Upper bound of the confidence interval = 0.04781 + (1.96 * 0.00215)
Lower bound of the confidence interval = 0.04781 - (1.96 * 0.00215)
These values will give you the 95% confidence interval for the true proportion of those from the entire mailing list who may donate.