Question

Suppose that you take an exam with 400 possible points and are told that the mean score is 280 and that the standard deviation is 20. you are also told that you got 350. Did you do well on the exam? Explain you answer.

Short answer

Yes, I did well on the exam.

By observing the three standard deviation rule, it implies that my possible given score is correct.

Step-by-step solution

Step 1. Given information.

Consider the given question,

The mean score is 280 and standard deviation is 20.

Maximum possible score is 400 and also told that you got score350.

Step 2. Find the z-score of score 350.

The z-score of score is given below,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{350-280}{20} \\ =3.5 \end{gathered}$$

On interpreting, we can say that the the score I got is 3.5 and standard deviation above the mean.

Step 3. To check if the exam score 400 is usual.

For unusual score we have to find one, two and three standard deviation of the mean.

Unusual observations lie between the ranges $\overline{x}\pm \sigma$,

$$\begin{gathered} =\left(\overline{x}-\sigma ,\overline{x}+\sigma \right) \\ =\left(280-20,280+20\right) \\ =\left(260,300\right) \end{gathered}$$

Unusual observations lie between the ranges $\overline{x}\pm 2\sigma$,

$$\begin{gathered} =\left(\overline{x}-2\sigma ,\overline{x}+2\sigma \right) \\ =\left(280-2\times 20,280+2\times 20\right) \\ =\left(240,320\right) \end{gathered}$$

Unusual observations lie between the ranges $\overline{x}\pm 3\sigma$,

$$\begin{gathered} =\left(\overline{x}-3\sigma ,\overline{x}+3\sigma \right) \\ =\left(280-3\times 20,280+3\times 20\right) \\ =\left(220,340\right) \end{gathered}$$

By observing the three standard deviation rule implies that my possible score is correct.