The opinions of all 500 students at a high school on whether they support Candidate A for student body president are shown below. Hide and shuffle all of the values. Take a random sample of 40 values from the population. Based on that sample, find the sample proportion and use that value to create a 95% confidence interval for the true proportion of the population who support Candidate A, rounding to the nearest thousandth.
Sample Yes: 31 Sample No: 9 Samples: 40
Confidence Interval for Population Proportion: (__ , __)
To find the confidence interval for the true proportion of the population who support Candidate A, we can use the formula:
Confidence interval = sample proportion ± critical value * standard error
First, let's calculate the sample proportion.
Sample proportion = (Number of Yes in the sample) / (Total number of samples)
= 31 / 40
= 0.775
Next, we need to determine the critical value. Since we want a 95% confidence interval, we can refer to the standard normal distribution table or a statistical software to find the critical value associated with a 95% confidence level. In this case, the critical value is approximately 1.96.
Now, let's calculate the standard error.
Standard error = sqrt((Sample proportion * (1 - Sample proportion)) / Sample size)
= sqrt((0.775 * (1 - 0.775)) / 40)
= sqrt(0.18303125 / 40)
= sqrt(0.00457578125)
≈ 0.0676
Finally, we can plug these values into the formula to calculate the confidence interval.
Confidence interval = 0.775 ± (1.96 * 0.0676)
= 0.775 ± 0.132896
≈ (0.642, 0.908)
Therefore, the 95% confidence interval for the true proportion of the population who support Candidate A is approximately (0.642, 0.908), rounded to the nearest thousandth.