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Out of 300 people sampled, 195 preferred Candidate A.

Round to three decimals.

Based on this estimate, what proportion (as a decimal) of the voting population ( [tex]$p$[/tex] ) prefers Candidate A?

[tex]$\square$[/tex]

Compute a [tex]$95\%$[/tex] confidence interval, and give your answers to 3 decimal places.

[tex]$\square$[/tex] [tex]$\ \textless \ p \ \textless \ $[/tex] [tex]$\square$[/tex]



Answer :

GinnyAnswer
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]

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