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
To find the sample proportion and create a confidence interval, we need to follow these steps:
Step 1: Calculate the sample proportion.
The sample proportion (p̂) is calculated by dividing the number of "Yes" responses by the total sample size.
In this case, the sample proportion is calculated as:
p̂ = (Number of "Yes" responses) / (Total Sample Size)
= 31 / 40
= 0.775
Step 2: Calculate the standard error.
The standard error (SE) is a measure of the variability in the sample proportion and is calculated as:
SE = √[(p̂ * (1 - p̂)) / n]
where p̂ is the sample proportion, and n is the sample size.
In this case, the standard error is calculated as:
SE = √[(0.775 * (1 - 0.775)) / 40]
= √[(0.775 * 0.225) / 40]
= √(0.0174375 / 40)
= √0.0004359375
= 0.020873 (rounded to the nearest thousandth)
Step 3: Calculate the margin of error.
The margin of error is determined by multiplying the standard error by the critical value. The critical value is obtained from the standard normal distribution and is based on the desired confidence level.
For a 95% confidence level, the critical value (Z) is approximately 1.96.
The margin of error (ME) is calculated as:
ME = Z * SE
= 1.96 * 0.020873
= 0.040803 (rounded to the nearest thousandth)
Step 4: Calculate the confidence interval.
The confidence interval represents the range within which we can be confident that the true proportion of the population lies.
The confidence interval is calculated by subtracting the margin of error from the sample proportion (lower bound) and adding the margin of error to the sample proportion (upper bound).
In this case, the confidence interval is calculated as:
Lower Bound = p̂ - ME
= 0.775 - 0.040803
= 0.734197 (rounded to the nearest thousandth)
Upper Bound = p̂ + ME
= 0.775 + 0.040803
= 0.815803 (rounded to the nearest thousandth)
Therefore, the 95% confidence interval for the true proportion of the population who support Candidate A is approximately 0.734 to 0.816, rounding to the nearest thousandth.