Question

Scores of an achievement test show that it follows a normal distribution. Its mean is 78 with a standard deviation of 8. Find the interval between the middle 80% of the scores lie.

Answer 1

The middle 80% implies that we are looking for the probability for 40% of the scores either side of the mean. The z-core related to 40% is

$$P(Z<z)=0.5-0.4=0.1$$

$$z=-1.28155$$

$$\dfrac{x-78}{8}=-1.28155$$

$$x=78-1.28155(8)$$

$$x=67.7476$$

$$\dfrac{x-78}{8}=1.28155$$

$$x=78+1.28155(8)$$

$$x=88.2524$$

The interval is $(67.7476, 88.2524).$