In a random sample of 100 residents from a certain area it follows that 35 were in favour of the construction of a nuclear power plant. Constructs a 95% confidence interval for the proportion of all residents in favour of this plant. Give the upper limit.
To construct a confidence interval for the proportion of all residents in favor of the construction of a nuclear power plant, we can use the formula for a confidence interval for a proportion:
Confidence Interval = (Sample Proportion) ± (Critical Value) * (Standard Error)
First, we need to calculate the sample proportion:
Sample Proportion = (Number of residents in favor of the construction) / (Sample Size)
Sample Proportion = 35 / 100
Sample Proportion = 0.35
The critical value for a 95% confidence interval can be found by using a z-table or a calculator, and it is typically approximately 1.96 for large sample sizes.
Standard Error = sqrt((Sample Proportion * (1 - Sample Proportion)) / (Sample Size))
Standard Error = sqrt((0.35 * (1 - 0.35)) / 100)
Standard Error = sqrt(0.35 * 0.65 / 100)
Standard Error = sqrt(0.2275 / 100)
Standard Error = sqrt(0.002275)
Standard Error ≈ 0.0477
Now, we can plug the values into the formula to calculate the confidence interval:
Confidence Interval = 0.35 ± 1.96 * 0.0477
Calculating the upper limit:
Confidence Interval (Upper Limit) = 0.35 + 1.96 * 0.0477
Confidence Interval (Upper Limit) = 0.35 + 0.0937
Confidence Interval (Upper Limit) ≈ 0.443
Therefore, the upper limit of the 95% confidence interval for the proportion of all residents in favor of the construction of a nuclear power plant is approximately 0.443.
To construct a 95% confidence interval for the proportion of all residents in favor of the nuclear power plant, you can use the formula:
Confidence interval = sample proportion ± (Z * √(p * (1 - p) / n))
Where:
- sample proportion (p̂) is the proportion in the sample in favor of the plant,
- Z is the critical value for the desired confidence level (in this case, 95% confidence level),
- n is the sample size.
First, calculate the sample proportion:
p̂ = 35 / 100 = 0.35
Next, find the critical value (Z) for a 95% confidence level. The critical value corresponds to the area in the tails of the standard normal distribution. For a 95% confidence level, the critical value is approximately 1.96.
Using the formula and substituting the values:
Confidence interval = 0.35 ± (1.96 * √(0.35 * (1 - 0.35) / 100))
Now, calculate the values within the confidence interval:
Confidence interval = 0.35 ± (1.96 * √(0.35 * 0.65 / 100))
Confidence interval = 0.35 ± (1.96 * √(0.2275 / 100))
Confidence interval = 0.35 ± (1.96 * 0.0477)
Confidence interval = 0.35 ± 0.0936
Finally, calculate the upper limit of the confidence interval by adding the margin of error to the sample proportion:
Upper limit = 0.35 + 0.0936
Upper limit = 0.4436
Therefore, the upper limit of the 95% confidence interval for the proportion of all residents in favor of the nuclear power plant is approximately 0.4436.