- We verify the normal approximation criteria (\(np \ge 10\) and \(n(1-p) \ge 10\)).
- We determine the mean \(\mu = 48.4\) and standard deviation \(\sigma \approx 6.1443\).
- We apply the continuity correction to find the z-score for \(X > 38\), which is \(z = -1.61\).
- We compute the final probability \(P(Z > -1.61) = \boxed{0.9463}\).
Explanation
- Verify conditions and calculate parameters
To determine the probability that more than 38 individuals in a sample of 220 are single, we first verify if we can use the normal approximation to the binomial distribution. We are given the population proportion \(p = 0.22\) and sample size \(n = 220\). We calculate:
- Mean: \(\mu = np = 220 \times 0.22 = 48.4\)
- Standard deviation: \(\sigma = \sqrt{np(1-p)} = \sqrt{220 \times 0.22 \times 0.78} = \sqrt{37.752} \approx 6.1443\)
Since both \(np = 48.4\) and \(n(1-p) = 171.6\) are greater than or equal to 10, the normal approximation is appropriate.
- Apply continuity correction and calculate the z-score
Since we are using the normal distribution to approximate a discrete binomial distribution, we must apply a continuity correction. To calculate the probability of having more than 38 singles (\(X > 38\)), we adjust this boundary to \(X > 38.5\). We now calculate the corresponding z-score:
\[z = \frac{X - \mu}{\sigma} = \frac{38.5 - 48.4}{6.1443} = \frac{-9.9}{6.1443} \approx -1.61\]
Rounding to two decimal places as requested, we obtain \(z = -1.61\).
- Find the final probability
We now find the probability \(P(Z > -1.61)\) using the standard normal distribution table.
- The cumulative probability \(P(Z \le -1.61)\) is approximately \(0.0537\).
- Therefore, the probability of being greater than this value is \(P(Z > -1.61) = 1 - P(Z \le -1.61) = 1 - 0.0537 = 0.9463\).
The final probability is \(\boxed{0.9463}\).
Examples
In social science research, these calculations are used to understand population trends. For example, if a researcher estimates the proportion of a specific demographic, they can calculate the likelihood of observing a certain number of individuals in a random survey sample. This helps in validating hypotheses and determining if a sample result is statistically significant or expected by random chance. Mathematical expressions like \(z = \frac{\bar{x} - \mu}{\sigma}\) are fundamental to making these inferences.