In a sample of 400 voters, 360 indicated they favor the incumbent governor. The 95% confidence interval? I can't figure out what to do without a standard deviation!!!!!!!!

0.071 to 0.128

To calculate the 95% confidence interval for a proportion, we can use the formula:

CI = p̂ ± Z * √((p̂ * (1 - p̂)) / n)

Where:
- CI is the confidence interval
- p̂ is the sample proportion
- Z is the z-score corresponding to the desired confidence level
- n is the sample size

In this case, we are given that in a sample of 400 voters, 360 indicated they favor the incumbent governor. To find the sample proportion (p̂), we divide the number of voters who favor the governor by the total sample size:

p̂ = 360 / 400 = 0.9

The standard deviation (σ) is not directly given, but we can still calculate the confidence interval by using the Z-score. The Z-score corresponds to the desired confidence level and takes into account the sample size.

As you mentioned, the standard deviation is not provided. However, when dealing with proportions, we can assume the binomial distribution approaches a normal distribution when the sample size is large enough. Usually, when n * p̂ > 10 and n * (1 - p̂) > 10, we consider the sample size to be large enough.

In this case, n * p̂ = 400 * 0.9 = 360 and n * (1 - p̂) = 400 * 0.1 = 40, both of which are greater than 10. Thus, we can proceed with using the Z-score for the confidence interval calculation.

For a 95% confidence interval, the corresponding Z-score can be obtained from a standard normal distribution table or using statistical software. The Z-score for a 95% confidence interval is approximately 1.96.

Now we can substitute the values into the formula:

CI = 0.9 ± 1.96 * √((0.9 * (1 - 0.9)) / 400)

Simplifying further:

CI = 0.9 ± 1.96 * √(0.09 / 400)

CI = 0.9 ± 1.96 * 0.015

CI = 0.9 ± 0.0294

Finally, the 95% confidence interval is:

CI = (0.8706, 0.9294)

Therefore, we can be 95% confident that the true proportion of voters who favor the incumbent governor lies between 0.8706 and 0.9294.

To calculate a confidence interval for a proportion, you can use the formula:

CI = p̂ ± z * sqrt((p̂ * (1 - p̂)) / n)

Where:
- CI is the confidence interval
- p̂ is the sample proportion (favoring the incumbent governor)
- z is the z-value corresponding to the desired confidence level
- n is the sample size

In this case, since you do not have the standard deviation, you can estimate it using the sample proportion.

First, calculate the sample proportion p̂ by dividing the number of voters in favor of the incumbent governor by the total sample size:

p̂ = 360 / 400 = 0.9

Next, determine the z-value corresponding to the desired confidence level. For a 95% confidence level, the z-value is approximately 1.96.

Now, plug these values into the formula:

CI = 0.9 ± 1.96 * sqrt((0.9 * (1 - 0.9)) / 400)

Simplify the equation:

CI = 0.9 ± 1.96 * sqrt((0.9 * 0.1) / 400)

CI = 0.9 ± 1.96 * sqrt(0.09 / 400)

CI = 0.9 ± 1.96 * sqrt(0.000225)

CI = 0.9 ± 1.96 * 0.015

Finally, calculate the confidence interval:

CI = (0.9 - 1.96 * 0.015, 0.9 + 1.96 * 0.015)

CI = (0.8706, 0.9294)

Therefore, the 95% confidence interval for the proportion of voters favoring the incumbent governor is approximately (0.8706, 0.9294).