Answer to Question #301845 in Statistics and Probability for secret

Question

On the Standford-Binet test, the mean IQ is 100. A class of 21 kindergarten pupils were tested with a resulting mean of 99 and a standard deviation of 6.25. What is the margin of error of the 90% confidence interval of the average IQ of the kindergarten pupils?

Answer 1

When the population standard deviation is unknown, the mean has a Student's t-distribution.

The critical value for $\alpha = 0.10$ and $df = n-1 = 20$ degrees of freedom is $t_c = z_{1-\alpha/2; n-1} = 1.724718.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{X}-t_c\times\dfrac{s}{\sqrt{n}}, \bar{X}+t_c\times\dfrac{s}{\sqrt{n}})$$

The margin of error is

$$t_c\times\dfrac{s}{\sqrt{n}}=1.724718\times\dfrac{6.25}{\sqrt{21}}\approx2.352277$$