Out of 100 students sampled, 70 of them said that they hoped to get married someday. With 99.7% confidence, what is the approximate percentage of the students in the population who hope to get married someday?
A. 51.7% to 88.3%
B. 65.4% to 74.6%
C. 56.3% to 83.7%
D. 60.8% to 79.2%
To calculate the approximate percentage of the students in the population who hope to get married someday with 99.7% confidence, we can use the formula for a confidence interval:
Margin of error = Z * (sqrt((p*(1-p))/n))
Where:
Z = 2.968 (corresponding to 99.7% confidence for a large sample)
p = 70/100 = 0.7 (proportion of sampled students who hope to get married)
n = 100 (sample size)
Margin of error = 2.968 * (sqrt((0.7*(1-0.7))/100))
Margin of error = 2.968 * (sqrt(0.21/100))
Margin of error = 2.968 * (sqrt(0.0021))
Margin of error = 2.968 * 0.04584
Margin of error = 0.134
Now, we can calculate the confidence interval:
Lower bound = 0.7 - 0.134 = 0.566 = 56.6%
Upper bound = 0.7 + 0.134 = 0.834 = 83.4%
Therefore, the approximate percentage of the students in the population who hope to get married someday with 99.7% confidence is between 56.6% and 83.4%, which is closest to option C.
So, the answer is:
C. 56.3% to 83.7%