A student wanted to construct a 95% confidence interval for the average test score of a particular class. This was a challenging task as the student had to take into account various factors such as the sample size, the standard deviation of the population, and the level of confidence required. The student was determined to ensure that the confidence interval was accurate and reliable, which would help in making informed decisions about the class’s performance.
The student began by collecting data from a random sample of students in the class. The data included the test scores of each student, which were then used to calculate the sample mean and standard deviation. The student recognized that the sample mean would serve as an estimate of the population mean, and the standard deviation would help in determining the margin of error.
Next, the student needed to determine the critical value for a 95% confidence interval. The critical value is a multiplier that is used to calculate the margin of error and is based on the desired level of confidence and the distribution of the data. For a 95% confidence interval, the critical value is approximately 1.96 for a normal distribution.
With the critical value in hand, the student was able to calculate the margin of error. The margin of error is the maximum amount by which the sample mean can differ from the true population mean. To calculate the margin of error, the student used the formula:
Margin of Error = Critical Value (Standard Deviation / Square Root of Sample Size)
After obtaining the margin of error, the student could now construct the confidence interval. The confidence interval is a range of values that is likely to contain the true population mean. To calculate the confidence interval, the student used the formula:
Confidence Interval = Sample Mean ± Margin of Error
The student was careful to round the confidence interval to an appropriate number of decimal places to ensure accuracy. Once the confidence interval was calculated, the student could interpret the results and provide insights into the class’s performance.
In conclusion, the student successfully constructed a 95% confidence interval for the average test score of the class. This process involved collecting data, calculating the sample mean and standard deviation, determining the critical value, calculating the margin of error, and finally constructing the confidence interval. The student’s hard work and attention to detail resulted in a reliable and accurate confidence interval, which can now be used to make informed decisions about the class’s performance.
