[3] Question 8 A study is conducted to determine the mean age of students who take STTN111. A sample of 50 randomly chosen students from the 1200 students that take STTN111 showed a mean age of 19 with a standard deviation 2.8. If the mean age of all STTN111 students must be calculated accurately to the nearest year with \( 95 \% \) certainty, what is the smallest possible sample size that needs to be drawn?
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To calculate the necessary sample size for estimating the mean age with 95% certainty, you can use the formula for the required sample size \( n \) when estimating the mean: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] where: - \( Z \) is the z-score for the desired confidence level (1.96 for 95% confidence), - \( \sigma \) is the known population standard deviation (2.8 years), - \( E \) is the margin of error (1 year in this case, as you want to round to the nearest year). Plugging in the values gives: \[ n = \left( \frac{1.96 \cdot 2.8}{1} \right)^2 \] Calculating this, we find: \[ n = (5.488)^2 \approx 30.19 \] Since you can't take a fraction of a sample size, you would round up to 31. Therefore, you'd need a minimum sample size of 31 students to estimate the mean age with the desired level of confidence. Moreover, this process highlights the importance of sample size in research. Larger sample sizes generally yield more accurate and reliable results, minimizing the margin of error and ensuring better representation of the population. Fun fact: the concept of the margin of error can be likened to a game of darts—aiming for the bullseye (true population mean) means you want your darts (sample estimates) as close as possible, and a bigger dartboard (larger sample size) gives you a better chance!
