Answer to Question #301847 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 108 and a standard deviation of 7.09. What is the lower bound 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:

$$CI=(\bar{X}-t_c\times\dfrac{s}{\sqrt{n}}, \bar{X}+t_c\times\dfrac{s}{\sqrt{n}})$$ $$=(108-1.724718\times\dfrac{7.09}{\sqrt{21}},$$

$$108+1.724718\times\dfrac{7.09}{\sqrt{21}})$$

$$=(105.3318,110.6684)$$

The lower bound of the 90% confidence interval of the average IQ of the kindergarten pupils is 105.3318.