To solve the problems, we need to understand that we're dealing with a binomial distribution since there are two outcomes (having hypertension or not) for each individual in our sample.
a) Mean Number of Adults with Hypertension
In a binomial distribution, the mean (μ) can be calculated using the formula:
\[ \mu = n \cdot p \]
where:
- \( n \) is the number of trials (in this case, the number of adults in the sample, which is 20), and
- \( p \) is the probability of success (in this case, the proportion of adults with hypertension, which is 0.30).
Plugging in the values:
\[ \mu = 20 \cdot 0.30 = 6 \]
Thus, the mean number of adults who have hypertension is 6.00 (rounded to two decimal places).
b) Standard Deviation of the Number of Adults with Hypertension
The standard deviation (σ) for a binomial distribution can be calculated using the formula:
\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \]
Substituting \( n = 20 \) and \( p = 0.30 \):
\[ \sigma = \sqrt{20 \cdot 0.30 \cdot (1 - 0.30)} \] \[ \sigma = \sqrt{20 \cdot 0.30 \cdot 0.70} \] \[ \sigma = \sqrt{20 \cdot 0.21} \] \[ \sigma = \sqrt{4.2} \] \[ \sigma \approx 2.0494 \]
Thus, the standard deviation of the number of adults who have hypertension is 2.0494 (rounded to four decimal places).
In summary:
- The mean number of adults who have hypertension is 6.00.
- The standard deviation of the number of adults who have hypertension is 2.0494.
1 answer