To calculate the 90% confidence interval for the population parameter, we can use the formula:
\text{Confidence Interval} = \text{Sample Proportion} \pm \text{Critical Value} \times \text{Standard Error}
where:
- Sample Proportion is the proportion of voter support found in the poll (p̂)
- Critical Value is the value from the standard normal distribution corresponding to the desired confidence level (Z)
- Standard Error is the standard deviation of the sample proportion (σp̂)
Step 1: Calculate the Sample Proportion (p̂)
Given that the proportion of voters supporting the Green candidate is 25%, we can say that p̂ = 0.25.
Step 2: Calculate the Critical Value (Z)
Since we want a 90% confidence interval, we need to find the critical value that corresponds to a 5% margin of error on each side of the mean. Using a standard normal distribution table or calculator, we find that the critical value Z for a 90% confidence level is approximately 1.645.
Step 3: Calculate the Standard Error (σp̂)
The standard error of the sample proportion can be calculated using the formula:
\sigma_{p̂} = \sqrt{\frac{p̂(1-p̂)}{n}}
where n is the sample size. In this case, the sample size is 60.
\sigma_{p̂} = \sqrt{\frac{0.25(1-0.25)}{60}}
\sigma_{p̂} = \sqrt{\frac{0.1875}{60}}
\sigma_{p̂}\approx0.0559
Step 4: Calculate the Confidence Interval
Now we can plug in the values into the confidence interval formula:
\text{Confidence Interval}=0.25\pm1.645\times0.0559
Step 5: Calculate the Lower Limit
\text{Lower Limit}=0.25-1.645\times0.0559=0.25-0.0919555=0.1580445\approx15.80\%
Step 6: Calculate the Upper Limit
\text{Upper Limit}=0.25+1.645\times0.0559=0.25+0.0919555=0.3419555\approx34.20\%
Answer:
The lower limit of the 90% confidence interval is approximately 15.80%
The upper limit of the 90% confidence interval is approximately 34.20%