Math Snap

STEP 1

1. The true population proportion is $p = 0.85$.
2. The sample proportion is $\widehat{p} = 0.84$.
3. The sample size is $n = 115$.
4. We are using the normal approximation to construct confidence intervals.

STEP 2

1. Calculate the standard error of the sample proportion.
2. Compute the 75% confidence interval.
3. Compute the 90% confidence interval.

STEP 3

Calculate the standard error (SE) of the sample proportion using the formula:
$$SE = \sqrt{\frac{\widehat{p}(1 - \widehat{p})}{n}}$$ Substitute the given values:
$$SE = \sqrt{\frac{0.84 \times (1 - 0.84)}{115}}$$ $$SE = \sqrt{\frac{0.84 \times 0.16}{115}}$$ $$SE = \sqrt{\frac{0.1344}{115}}$$ $$SE = \sqrt{0.0011695652}$$ $$SE \approx 0.0342$$

STEP 4

Compute the 75% confidence interval using the critical value 1.150:
Lower limit:
$$\widehat{p} - (1.150 \times SE) = 0.84 - (1.150 \times 0.0342)$$ $$= 0.84 - 0.0393$$ $$= 0.8007$$ Upper limit:
$$\widehat{p} + (1.150 \times SE) = 0.84 + (1.150 \times 0.0342)$$ $$= 0.84 + 0.0393$$ $$= 0.8793$$ The 75% confidence interval is $[0.80, 0.88]$.

Compute the 90% confidence interval using the critical value 1.645:
Lower limit:
$$\widehat{p} - (1.645 \times SE) = 0.84 - (1.645 \times 0.0342)$$ $$= 0.84 - 0.0563$$ $$= 0.7837$$ Upper limit:
$$\widehat{p} + (1.645 \times SE) = 0.84 + (1.645 \times 0.0342)$$ $$= 0.84 + 0.0563$$ $$= 0.8963$$ The 90% confidence interval is $[0.78, 0.90]$.
The confidence intervals are:
- 75% confidence interval: $[0.80, 0.88]$
- 90% confidence interval: $[0.78, 0.90]$