### How to find the 95% confidence interval of the mean math test scores

A sample of the math test scores of 35 fourth-graders has a mean of 82 with a standard deviation of 15. (a) Find the 95% confidence interval of the mean math test scores of all fourth-graders. (b) Find the 99% confidence interval of the mean math test scores of all fourth-graders. (c) Which interval is larger? Explain why.

(a) Find the 95% confidence interval of the mean math test scores of all fourth-graders.

SOLUTION:

The 95% confidence interval of the mean math test scores of all fourth graders given that $$n=35, \bar{x}=82.0 \text{ and } \sigma =15.0$$ is
$\begin{array} {lcl}\bar{x}-z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}}< \mu< \bar{x}-z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}} &&=&& 82.0-1.96\dfrac{15.0}{\sqrt{35}}< 82.0-1.96\dfrac{15.0}{\sqrt{35}}\\&&=&&-77.03< \mu<86.97 \end{array}$ (b) Find the 99% confidence interval of the mean math test scores of all fourth-graders.

SOLUTION:

The 99% confidence interval of the mean math test scores of all fourth graders given that $$n=35, \bar{x}=82.0 \text{ and } \sigma =15.0$$ is
$\begin{array} {lcl}\bar{x}-z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}}< \mu< \bar{x}-z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}} &&=&& 82.0-2.575\dfrac{15.0}{\sqrt{35}}< 82.0-2.575\dfrac{15.0}{\sqrt{35}}\\&&=&&75.47< \mu< 88.53 \end{array}$ (c) Which interval is larger? Explain why.

ANSWER: The 99% confidence interval has a larger interval compared to the 95% confidence interval because 99% confidence interval has almost had the numbers in it. Only 1% is not included in the interval. If we have to be more confident one should pick or choose nearly the entire sample or even the entire sample. It means to say that the higher confident are we the greater the range of numbers we have.

For instance, consider a multiple choice question with 5 answers only one right and somehow you were allowed to pick all the options. How confident would you be one of the answers you picked would be the right answer? 100% confident because you picked everything! But now say someone says you can only pick 4. You must be less confident right? And 3 and 2 and 1? Each time you pick less answers no matter how sure you are you know the actual answers you are picking a smaller subset so your confidence that one of the things you picked covers the right answer has to be going down. The only 100% confidence interval has every number in it. A single point covers the truth with no confidence. And everything else is between those extremes. Just ask yourself is one of the intervals smaller than the other? Is the smaller interval wholly contained in the larger interval? If so how could you possibly be more confident that the smaller interval covered the actual answer when the larger interval has the entire smaller interval plus some more "stuff".