Math Snap

STEP 1

1. The sample size $n = 315$.
2. The number of successes $x = 185$.
3. We are using a $95\%$ confidence interval.
4. The population is large enough for the Central Limit Theorem to apply.

STEP 2

1. Calculate the sample proportion $\hat{p}$.
2. Calculate the standard error estimate $\mathrm{SE}$.
3. Determine the critical value $z$ for a $95\%$ confidence interval.
4. Calculate the margin of error $m$.

STEP 3

Calculate the sample proportion $\hat{p}$:
$$\hat{p} = \frac{x}{n} = \frac{185}{315}$$ $$\hat{p} \approx 0.587$$

STEP 4

Calculate the standard error estimate $\mathrm{SE}$:
$$\mathrm{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.587 \times (1 - 0.587)}{315}}$$ $$\mathrm{SE} \approx 0.028$$

STEP 5

Determine the critical value $z$ for a $95\%$ confidence interval. Using a $z$-distribution table, the critical value is:
$$z \approx 1.96$$

Calculate the margin of error $m$:
$$m = z \times \mathrm{SE} = 1.96 \times 0.028$$ $$m \approx 0.055$$ The sample proportion is:
$$\hat{p} = 0.587$$ The standard error estimate is:
$$\mathrm{SE} = 0.028$$ The critical value is:
$$z = 1.96$$ The margin of error is:
$$m = 0.055$$