To solve these questions, we can utilize the properties of a binomial distribution. Here, the number of trials \( n \) is 27 (the number of high school graduates sampled), and the probability of success (enrolling in college) \( p \) is 0.64.
(a) Mean Number Who Enroll in College
The mean \( \mu \) of a binomial distribution is calculated using the formula:
\[ \mu = n \cdot p \]
Substituting the values:
\[ \mu = 27 \cdot 0.64 = 17.28 \]
Thus, the mean number of students who enroll in college is 17.28 (rounded to two decimal places).
(b) Standard Deviation of the Number Who Enroll in College
The standard deviation \( \sigma \) of a binomial distribution is calculated using the formula:
\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \]
First, we need to calculate \( (1 - p) \):
\[ 1 - p = 1 - 0.64 = 0.36 \]
Now substituting the values into the formula for standard deviation:
\[ \sigma = \sqrt{27 \cdot 0.64 \cdot 0.36} \]
Calculating the inside of the square root first:
\[ 27 \cdot 0.64 = 17.28 \] \[ 17.28 \cdot 0.36 = 6.2208 \]
Now, taking the square root:
\[ \sigma = \sqrt{6.2208} \approx 2.4942 \]
Thus, the standard deviation of the number of students who enroll in college is approximately 2.4942 (rounded to four decimal places).
Summary:
- (a) Mean: 17.28
- (b) Standard Deviation: 2.4942