A simple random sample of 1350 registered voters shows that 58% favor Candidate A over Candidate B.
Construct a 99% confidence interval for the percent of the population of all registered voters who prefer Candidate A over Candidate B.
To construct a confidence interval for the percent of the population who prefer Candidate A over Candidate B, we can use the following formula:
Confidence Interval = sample proportion ± (critical value) * (standard deviation)
Let's break down the steps to calculate this confidence interval:
Step 1: Identify the sample proportion (p̂)
The sample proportion is the percent of the sample who prefer Candidate A over Candidate B. In this case, it is 58%, which can be expressed as 0.58 (since it is a decimal).
Step 2: Determine the critical value
The critical value is based on the desired confidence level. For a 99% confidence level, we need to find the critical value associated with a 99% confidence level. We can use a Z-table or a calculator to find this value. For a 99% confidence level, the critical value is approximately 2.576.
Step 3: Calculate the standard deviation (SD)
The standard deviation measures the variability in the data. In this case, we need to calculate the standard deviation of the sample proportion using the formula:
SD = √ [(p̂ * (1 - p̂)) / n]
where p̂ is the sample proportion, and n is the sample size. In this case, p̂ = 0.58 and n = 1350.
Step 4: Plug in the values and calculate the confidence interval
Using the formula mentioned earlier, we can calculate the confidence interval:
Confidence Interval = p̂ ± (critical value) * (SD)
Confidence Interval = 0.58 ± 2.576 * (√ [(0.58 * (1 - 0.58)) / 1350])
Now, you can simplify the expression and solve for the lower and upper bounds of the confidence interval to get the final result.
Please note that the confidence interval reflects our level of confidence that the true population proportion lies within the interval. In this case, the confidence level is 99%, meaning that if we repeat this process many times, we would expect the true population proportion to fall within this range 99% of the time.