Thirty five percent of adult americans are regular voters. A random sample of 250 adults in a medium size college town were surveyed, and it was found that 110 were regular voters. Estimate the true proportion of regular voters with 90% confidence and comment on the results. Any help would be greatly appreciated. Please include the steps so I can understand how to do these problems.
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To estimate the true proportion of regular voters with 90% confidence, you can use the concept of confidence intervals. The formula for calculating a confidence interval for a proportion is given by:
Confidence Interval = (sample proportion) ± (critical value) * (standard error)
To estimate the true proportion of regular voters, follow these steps:
Step 1: Calculate the sample proportion:
In this case, the sample proportion is the number of regular voters in the sample divided by the total sample size. So, the sample proportion is calculated as:
sample proportion = (number of regular voters / sample size) = 110 / 250 = 0.44
Step 2: Calculate the critical value:
The critical value corresponds to the desired confidence level, which in this case is 90%. To find the critical value, you need to consult a table of critical values for the desired confidence level. For a 90% confidence level, the critical value (z-score) is approximately 1.645.
Step 3: Calculate the standard error:
The standard error measures the variability or precision of the sample proportion. It can be calculated using the formula:
standard error = sqrt((sample proportion * (1 - sample proportion)) / sample size)
standard error = sqrt((0.44 * (1 - 0.44)) / 250) ≈ 0.038
Step 4: Calculate the confidence interval:
Now that you have the sample proportion, critical value, and standard error, you can calculate the confidence interval using the formula mentioned earlier:
Confidence Interval = sample proportion ± (critical value) * (standard error)
Confidence Interval = 0.44 ± 1.645 * 0.038
Calculating the upper and lower limits of the confidence interval:
Lower Limit = 0.44 - (1.645 * 0.038) ≈ 0.375
Upper Limit = 0.44 + (1.645 * 0.038) ≈ 0.505
The confidence interval is approximately 0.375 to 0.505.
Step 5: Interpretation:
The 90% confidence interval (0.375 to 0.505) means that we can be 90% confident that the true proportion of regular voters in the population lies between 0.375 and 0.505. In other words, based on this sample, with 90% confidence, we estimate that the proportion of regular voters in the population may range from 37.5% to 50.5%.
Comment on the results:
Since the interval does not include the value 0.35 (the provided estimate for the proportion of regular voters in the population), it suggests that the college town may have a higher proportion of regular voters than the national average. However, this is just an estimation, and it is important to keep in mind the margin of error and the probability that the true proportion falls outside the interval.