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. Contruct a 95% confidence interval for the proportion of all residents in favour of this plant.

1 answer

To construct a 95% confidence interval for the proportion of residents in favor of the construction of a nuclear power plant, we can use the formula for the confidence interval for a population proportion.

Step 1: Calculate the sample proportion

The sample proportion \(\hat{p}\) is given by:

\[ \hat{p} = \frac{x}{n} \]

where \(x\) is the number of successes (in favor) and \(n\) is the sample size.

From the problem:

  • \(x = 35\)
  • \(n = 100\)

So,

\[ \hat{p} = \frac{35}{100} = 0.35 \]

Step 2: Calculate the standard error

The standard error (SE) for the proportion is calculated using the formula:

\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Substitute \(\hat{p} = 0.35\) and \(n = 100\):

\[ SE = \sqrt{\frac{0.35 \times (1 - 0.35)}{100}} = \sqrt{\frac{0.35 \times 0.65}{100}} = \sqrt{\frac{0.2275}{100}} = \sqrt{0.002275} \approx 0.0477 \]

Step 3: Find the Z-score for 95% confidence

For a 95% confidence level, the Z-score (critical value) is approximately 1.96.

Step 4: Calculate the margin of error

The margin of error (ME) is calculated as follows:

\[ ME = Z \times SE \]

So,

\[ ME = 1.96 \times 0.0477 \approx 0.0935 \]

Step 5: Construct the confidence interval

The confidence interval is given by:

\[ \hat{p} \pm ME \]

Calculating the lower and upper bounds:

  • Lower bound: \(0.35 - 0.0935 \approx 0.2565\)
  • Upper bound: \(0.35 + 0.0935 \approx 0.4435\)

Final Result

Thus, the 95% confidence interval for the proportion of all residents in favor of the nuclear power plant is approximately:

\[ (0.2565, 0.4435) \]

This means we are 95% confident that the true proportion of all residents in favor of the construction of the nuclear power plant lies within the interval \(0.2565\) to \(0.4435\).