In a random sample of 200 registered voters, 150 indicated they are Democrats. Develop a 99% confidence interval for the proportion of registered voters in the population who are Democrats.
try poking around here, if you don't have a handy Z table:
http://davidmlane.com/hyperstat/z_table.html
To develop a confidence interval for the proportion of registered voters in the population who are Democrats, we can use the formula for constructing a confidence interval for a proportion. The formula takes into account the sample proportion, the sample size, and the desired level of confidence.
Let's start by identifying the key information given in the question:
Sample Size (n): 200
Number of Democrats in the sample (x): 150
Level of Confidence: 99%
Now, we can calculate the confidence interval using the following steps:
Step 1: Calculate the sample proportion (p-hat).
The sample proportion is obtained by dividing the number of Democrats in the sample (x) by the sample size (n).
p-hat = x / n
p-hat = 150 / 200
p-hat = 0.75
Step 2: Determine the critical value (Z-value).
The critical value is dependent on the desired level of confidence. For a 99% confidence interval, the critical value corresponds to the area outside the confidence interval, which can be retrieved from a standard normal distribution table or calculator.
Since we need a two-tailed confidence interval, we will divide the significance level (1 - confidence level) by 2 and find the corresponding Z-value. In this case, (1 - 0.99) / 2 = 0.005.
Using a standard normal distribution table or calculator, the Z-value for a significance level of 0.005 is approximately 2.576.
Step 3: Calculate the margin of error (E).
The margin of error represents the distance from the sample proportion to the upper and lower bounds of the confidence interval. It is given by the product of the critical value (Z-value) and the standard deviation of the sampling distribution (which can be approximated using the formula sqrt((p-hat * (1 - p-hat)) / n)).
E = Z * sqrt((p-hat * (1 - p-hat)) / n)
E = 2.576 * sqrt((0.75 * (1 - 0.75)) / 200)
E ≈ 0.084
Step 4: Calculate the lower and upper bounds of the confidence interval.
The lower bound is obtained by subtracting the margin of error from the sample proportion (p-hat), while the upper bound is obtained by adding the margin of error to the sample proportion.
Lower Bound = p-hat - E
Lower Bound = 0.75 - 0.084
Lower Bound ≈ 0.666
Upper Bound = p-hat + E
Upper Bound = 0.75 + 0.084
Upper Bound ≈ 0.834
Therefore, the 99% confidence interval for the proportion of registered voters in the population who are Democrats is approximately 0.666 to 0.834.