Question

Five students visiting the student health center for a free dental examination during National Dental Hygiene Month were asked how many months had passed since their last visit to a dentist. Their responses were as follows: \(\begin{array}{lllll}6 & 17 & 11 & 22 & 29\end{array}\) Assuming that these five students can be considered a random sample of all students participating in the free checkup program, construct a \(95 \%\) confidence interval for the mean number of months elapsed since the last visit to a dentist for the population of students participating in the program.

Short answer

The solution will involve applying statistical concepts related to calculating the sample mean, the sample standard deviation, the standard error, finding corresponding critical value and finally calculating the 95% confidence interval. The final result will be an interval that estimates the average number of months since the last dentist visit for the population of students participating in the free checkup program.

Step-by-step solution

Calculating the sample mean

First, calculate the mean (average) of the provided data. This can be done by adding up the months recorded for all five students and then dividing by the number of students. In this case, the calculation is as follows: (6+17+11+22+29)/5.

Calculating the standard error

Next, calculate the standard error of the mean. This is the standard deviation of the sample divided by the square root of the sample size. Since this is a small sample and we don't know the population standard deviation, we must first calculate the sample standard deviation using the formula for the sample standard deviation. Then proceed with the calculation of the standard error.

Finding the critical value

Then, find the critical value that corresponds to a 95% confidence level. Since the sample size is small and the population standard deviation is unknown, use the t-distribution. For a 95% confidence level and 4 degrees of freedom (since n-1=5-1), the critical value (also called t-star) is commonly 2.776.

Constructing the confidence interval

The formula for a confidence interval is: Confidence Interval = sample mean ± (critical value * standard error). Substitute the values previously calculated into this formula to obtain the 95% confidence interval.