A high school counselor is interested in the percentage of students who will be going
to college. She randomly picks 60 students and finds that 53 will be going to college.
Use a 95% confidence level to find the population proportion.
a) State the 95% confidence interval for population proportion.
(Round to tenth of a percent)
b) State the margin of error (Round to tenth of a percent)
To find the 95% confidence interval for the population proportion, we can use the formula:
Confidence Interval = Sample proportion ± Margin of Error
First, let's calculate the sample proportion using the data given. The proportion of students going to college is determined by dividing the number of students going to college by the total number of students in the sample.
Sample proportion = Number of students going to college / Total number of students in the sample = 53 / 60 = 0.8833 (rounded to four decimal places)
To determine the margin of error, we'll use the formula for a sample proportion:
Margin of Error = Critical value * Standard Error
For a 95% confidence level, the critical value is obtained from a Z-table, and it is approximately 1.96.
Standard Error = sqrt((Sample proportion * (1 - Sample proportion)) / Sample size)
Sample size = Total number of students in the sample = 60
Now, let's calculate the standard error:
Standard Error = sqrt((0.8833 * (1 - 0.8833)) / 60) ≈ 0.0415 (rounded to four decimal places)
Next, we can calculate the margin of error:
Margin of Error = 1.96 * 0.0415 ≈ 0.0812 (rounded to four decimal places)
Finally, we can calculate the confidence interval by subtracting and adding the margin of error to the sample proportion:
Confidence Interval = 0.8833 ± 0.0812 (rounded to four decimal places)
a) The 95% confidence interval for the population proportion is approximately 0.8021 to 0.9645 (rounded to the tenth of a percent).
b) The margin of error is approximately 0.0812 (rounded to the tenth of a percent).