in a random sample of 400 voter, 120 plan to vote for candidate A. determine a 95% confidence interval
To determine a 95% confidence interval for the proportion of voters planning to vote for candidate A, we can use the formula:
Confidence interval = sample proportion ± critical value * standard error
1. Calculate the sample proportion:
sample proportion = number of voters planning to vote for candidate A / total sample size
= 120 / 400
= 0.3
2. Determine the critical value for a 95% confidence interval. For a 95% confidence interval, the critical value is 1.96. This value corresponds to the desired level of confidence (95%) and is based on the standard normal distribution.
3. Calculate the standard error:
standard error = √[(sample proportion * (1 - sample proportion)) / sample size]
= √[(0.3 * (1 - 0.3)) / 400]
= √[(0.21) / 400]
= √(0.000525)
≈ 0.022911
4. Finally, calculate the confidence interval:
Confidence interval = 0.3 ± (1.96 * 0.022911)
= 0.3 ± 0.044938
≈ (0.255, 0.345)
Therefore, with 95% confidence, the proportion of voters planning to vote for candidate A is estimated to be between 25.5% and 34.5%.
To determine a 95% confidence interval, we need to know the sample proportion and the sample size.
In this case, the sample size is 400 voters, and 120 of them plan to vote for candidate A. The sample proportion (p-hat) can be calculated by dividing the number of voters planning to vote for candidate A by the total sample size:
p-hat = 120/400 = 0.3
Now, we can calculate the standard error using the following formula:
SE = √ (p-hat * (1 - p-hat) / n)
Where p-hat is the sample proportion and n is the sample size.
SE = √ (0.3 * (1 - 0.3) / 400)
= √ (0.3 * 0.7 / 400)
≈ 0.0218
The 95% confidence interval can then be calculated as:
CI = p-hat ± (Z * SE)
Where Z is the value corresponding to the desired confidence level. For a 95% confidence level, Z is approximately 1.96 (for a large sample size).
CI = 0.3 ± (1.96 * 0.0218)
Therefore, the 95% confidence interval for the proportion of voters planning to vote for candidate A is approximately:
CI = (0.257, 0.343)