Question

On a nationwide test taken by high school students, the mean score was 47 and the standard deviation was 13. The scores were normally distributed. Complete the following statements. (a) Approximately ____ of the students scored between 34 and 60. (b) Approximately 99.7% of the students scored between __ and ___

Answer 1

Concept:

The question will be solved using the empirical rule below

The Empirical Rule states that 99.7% of data observed following a normal distribution lies within 3 standard deviations of the mean. Under this rule, 68% of the data falls within one standard deviation, 95% percent within two standard deviations, and 99.7% within three standard deviations from the mean.

a) To figure out the percentage of students that scored between 34 and 60, we will use the principle below

As the scores are normally distributed so, according to empirical rule the percentage of data falls within one,two and three standard deviations are 68%,95% and 99.7% respectively.

`\begin{gathered} 68\%=\mu-\sigma,\mu+\sigma \\ 95\%=\mu-2\sigma,\mu+2\sigma \\ 99.7\%=\mu-3\sigma,\mu+3\sigma \end{gathered}`

Where the mean and standard deviation given are

`\mu=47,\sigma=13`

By substituting the values, we will have

Hence,

Approximately __68%__ of the students scored between 34 and 60.

B)

Approximately 99.7% of the students scored between __ and ___​

To figure out the values, we will use the formula below

By substituting the values, we will have

Hence,

The final answer is

Approximately 99.7% of the students scored between _8_ and _86__​