Sure! Let's solve this problem step by step.
### Step 1: Calculate the Sample Proportion ([tex]\(\hat{p}\)[/tex])
Given:
- Total number of people sampled ([tex]\(n\)[/tex]): 300
- Number of people who preferred Candidate A ([tex]\(x\)[/tex]): 195
The sample proportion ([tex]\(\hat{p}\)[/tex]) is calculated as:
[tex]\[ \hat{p} = \frac{x}{n} \][/tex]
Plugging in the values:
[tex]\[ \hat{p} = \frac{195}{300} = 0.65 \][/tex]
### Step 2: Calculate the Standard Error (SE)
Standard error (SE) is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} \][/tex]
Plugging in the values (where [tex]\(\hat{p}\)[/tex] is 0.65 and [tex]\(n\)[/tex] is 300):
[tex]\[ SE = \sqrt{\frac{0.65 \cdot (1 - 0.65)}{300}} = \sqrt{\frac{0.65 \cdot 0.35}{300}} \][/tex]
[tex]\[ SE = \sqrt{\frac{0.2275}{300}} \approx 0.027 \][/tex]
### Step 3: Determine the Z-Value for a 95% Confidence Level
For a 95% confidence level, the z-value is typically 1.96.
### Step 4: Calculate the Confidence Interval
The confidence interval is given by:
[tex]\[ \hat{p} \pm (z \cdot SE) \][/tex]
Plugging in the values:
[tex]\[ \text{Lower bound} = \hat{p} - (z \cdot SE) = 0.65 - (1.96 \cdot 0.027) \approx 0.65 - 0.053 = 0.596 \][/tex]
[tex]\[ \text{Upper bound} = \hat{p} + (z \cdot SE) = 0.65 + (1.96 \cdot 0.027) \approx 0.65 + 0.053 = 0.704 \][/tex]
### Step 5: Present the Results
The proportion ([tex]\(\hat{p}\)[/tex]) of the voting population that prefers Candidate A is:
[tex]\[ 0.65 \][/tex]
The 95% confidence interval for the true proportion [tex]\(p\)[/tex] is:
[tex]\[ 0.596 < p < 0.704 \][/tex]
So, the final answers are:
Proportion:
[tex]\[ 0.65 \][/tex]
95% Confidence Interval:
[tex]\[ 0.596 < p < 0.704 \][/tex]
