when 325 students are randomly selected and surveyed, it is found that 117 own a car. Construct a 99% confidence interval estimate of the true proportion of all college students who own a car.
To construct a confidence interval estimate of the true proportion, you can use the formula:
Confidence Interval = Sample proportion ± (Z * Standard Error)
Where:
- Sample proportion is the number of students owning a car divided by the total number of students surveyed.
- Z is the Z-value corresponding to the desired level of confidence. For a 99% confidence level, the Z-value is approximately 2.576.
- Standard Error is the standard deviation of the sampling distribution, which can be calculated as the square root of [(Sample proportion * (1 - Sample proportion)) / n], where n is the sample size.
Now, let's calculate the confidence interval step-by-step:
1. Calculate the Sample proportion: p-hat = 117/325 = 0.36.
2. Calculate the Standard Error: SE = sqrt[(0.36 * (1 - 0.36)) / 325] ≈ 0.0306.
3. Determine the Z-value for a 99% confidence level: Z = 2.576.
4. Calculate the Lower Confidence Limit (LCL):
LCL = p-hat - (Z * SE) = 0.36 - (2.576 * 0.0306) ≈ 0.2783.
5. Calculate the Upper Confidence Limit (UCL):
UCL = p-hat + (Z * SE) = 0.36 + (2.576 * 0.0306) ≈ 0.4417.
Therefore, the 99% confidence interval estimate for the true proportion of all college students who own a car is approximately 0.2783 to 0.4417.