Sample Size(n):
Sample Mean(Ⅹ):
Standard Deviation(σ/s):
Confidence Level(%):
to
result
\(CL = \overline{X} \pm Z \times \frac{\sigma}{\sqrt{n}}\)
A confidence interval (CI) is a range used in statistics to estimate a population parameter. For example, suppose we have a set of data representing the exam scores of students in a class (in points): 85, 90, 88, 92, 87, 89, 91, 86, 88, 90. We want to calculate the confidence interval at a 95% confidence level for this data set. Through calculation, we find that the 95% confidence interval for this data set is [87.01, 90.18]. This means we are 95% confident that the true mean of all students' exam scores falls within this interval.
The confidence level is an indicator of the reliability of the confidence interval. Common confidence levels include 90%, 95%, and 99%, among others. For instance, a 95% confidence level means we are 95% confident that the true value of the population parameter lies within the calculated confidence interval. Simply put, it reflects the probability you require for the accuracy of the estimated result, also known as confidence degree. For example, with a requirement of 95% accuracy, the estimated exam scores for the class are [87.01, 90.18], and this 95% represents the confidence level.
We will use the exam scores of students in a certain class (in points): 85, 90, 88, 92, 87, 89, 91, 86, 88, 90 as an example to calculate the confidence interval.
Step1：We first calculate the sample mean, which is 88.6
\(\overline{X} = \frac{85 + 90 + 88 + 92 + 87 + 89 + 91 + 86 + 88 + 90}{10} = 88.6\)
Step2： Calculate the sample standard deviation s, which is 2.16
\(s = \sqrt{\frac{\sum_{i=1}^{n}(X_{i} - \overline{X})^{2}}{n - 1}} = 2.16\)
Step3：Determine the Confidence Level
Assume we use a 95% confidence level and find the corresponding Z critical value (for a 95% confidence level, the Z critical value is 1.96)
Step4：Calculate the Standard Error (SE)
\( SE = \frac{s}{\sqrt{n}} = \frac{2.16}{\sqrt{10}} \approx 0.68 \)
Step5：Calculate the Confidence Interval(CI)
\( Cl = \overline{X} \pm Z \times SE \)
\( Cl = 88.6 \pm 1.96 \times 0.68 \)
\( Cl = 88.6 \pm 1.33 \)
\( Cl = (87.27, 89.93)) \)
Result
Therefore, the 95% confidence interval for the given data is: (87.27,89.93)
