Let's go through the problem step-by-step and interpret the given results.
### Step 1: Given Data
1. Sample Size (n): 124 residents
2. Number Opposed (x): 93 residents
3. Desired Confidence Level: 90%, which corresponds to a [tex]\( z^ \)[/tex] score of 1.645
### Step 2: Calculate the Point Estimate of the Population Proportion ([tex]\(\hat{p}\)[/tex])
The point estimate ([tex]\(\hat{p}\)[/tex]) is calculated as:
[tex]\[ \hat{p} = \frac{x}{n} \][/tex]
Substituting the values:
[tex]\[ \hat{p} = \frac{93}{124} \][/tex]
[tex]\[ \hat{p} = 0.75 \][/tex]
### Step 3: Calculate the Margin of Error (E)
The margin of error (E) is calculated using the formula:
[tex]\[ E = z^ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \][/tex]
Substituting the values:
[tex]\[ E = 1.645 \times \sqrt{\frac{0.75(1 - 0.75)}{124}} \][/tex]
[tex]\[ E = 0.06396695766868311 \][/tex]
The margin of error in percentage terms is:
[tex]\[ E \times 100 = 6.396695766868311 \% \][/tex]
### Step 4: Addressing the Statements
1. The sample size is 93.
- This statement is false. The sample size is 124. The number 93 represents the number of residents opposed to the new shopping center within the sample.
2. The sample size is 310.
- This statement is false. The total population of the town is 310, but our sample size is 124.
3. The point estimate of the population proportion is 0.4.
- This statement is false. The correct point estimate is 0.75.
4. The point estimate of the population proportion is 0.75.
- This statement is true. As calculated, [tex]\(\hat{p} = 0.75\)[/tex].
5. The margin of error is approximately 4%.
- This statement is false. The margin of error is approximately 6.4%.
6. The margin of error is approximately 6%.
- This statement is true. The margin of error is approximately 6.4%, which rounds to 6%.
### Conclusion
Based on the given data and the detailed calculations, the following statements are true:
- The point estimate of the population proportion is 0.75.
- The margin of error is approximately 6%.
These results are consistent with the provided numerical values of [tex]\(\hat{p}\)[/tex] and the margin of error.
