To find the 95% confidence interval for the sample proportion, we can follow these steps:
Step 1: Calculate the sample proportion.
The sample proportion is calculated by dividing the number of voters who support candidate A (110) by the total number of voters in the sample (275):
Sample proportion (pĖ) = Number of voters supporting candidate A / Total number of voters
= 110 / 275
= 0.4
Step 2: Calculate the standard error.
The standard error represents the estimated standard deviation of the sample proportion. It can be calculated using the formula:
Standard error (SE) = â(pĖ(1 - pĖ) / n)
Where:
pĖ is the sample proportion
n is the sample size
SE = â(0.4(1 - 0.4) / 275)
SE â 0.026
Step 3: Calculate the margin of error.
The margin of error is a measure of the uncertainty in the sample proportion. It can be calculated by multiplying the standard error by the critical value. In this case, the critical value is z=1.960 (corresponding to a confidence level of 95%):
Margin of error = z * SE
= 1.960 * 0.026
â 0.051
Step 4: Determine the confidence interval.
The confidence interval is calculated by subtracting and adding the margin of error from the sample proportion:
Lower bound = pĖ - Margin of error
= 0.4 - 0.051
â 0.349
Upper bound = pĖ + Margin of error
= 0.4 + 0.051
â 0.451
So, the 95% confidence interval for the sample proportion that support candidate A is approximately 34.9% to 45.1%. Rounded to the nearest tenth of a percent, the confidence interval is 35% to 45%.