Question 9
A state representative wishes to estimate the mean
number of women representatives per state
legislature. A random sample of 17 was selected, and
from the sample, the point estimate of the mean is
33.4, and the variance is 823.69, find the
confidence interval of the mean population. (write the
lower limit for the confidence interval).
Add your answer
Integer, decimal, or E notation allowed

Question

Question 9
A state representative wishes to estimate the mean
number of women representatives per state
legislature. A random sample of 17 was selected, and
from the sample, the point estimate of the mean is
33.4, and the variance is 823.69, find the

confidence interval of the mean population. (write the
lower limit for the confidence interval).
Add your answer
Integer, decimal, or E notation allowed

UpStudy AI · Accepted Answer

To find the lower limit of the 90% confidence interval for the mean number of women representatives per state legislature, follow these steps:

### Given:
- **Sample size ($$ n $$)**: 17
- **Sample mean ($$ \bar{x} $$)**: 33.4
- **Sample variance ($$ s^2 $$)**: 823.69

### Steps:

1. **Calculate the Sample Standard Deviation ($$ s $$)**:
   $$
   s = \sqrt{823.69} = 28.7
   $$

2. **Determine the Standard Error (SE)**:
   $$
   SE = \frac{s}{\sqrt{n}} = \frac{28.7}{\sqrt{17}} \approx \frac{28.7}{4.1231} \approx 6.9752
   $$

3. **Find the t-Score for a 90% Confidence Interval with 16 Degrees of Freedom**:
   - For a 90% confidence level, $$ \alpha = 0.10 $$, so each tail has $$ \alpha/2 = 0.05 $$.
   - From the t-distribution table, $$ t_{0.05, 16} \approx 1.746 $$.

4. **Calculate the Margin of Error (ME)**:
   $$
   ME = t \times SE = 1.746 \times 6.9752 \approx 12.1785
   $$

5. **Determine the Lower Limit of the Confidence Interval**:
   $$
   \text{Lower Limit} = \bar{x} - ME = 33.4 - 12.1785 \approx 21.22
   $$

### **Answer:**
The lower limit of the 90% confidence interval is **21.22**.
The lower limit of the 90% confidence interval is 21.22.

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