Step 1

Empirical rule:

For symmetric or normal distributions,

68% of values fall within one standard deviation from the mean. That is, (m – s, m + s).

95% of values fall within two standard deviations from the mean. That is, (m – 2*s, m + 2*s).

7% of values fall within three standard deviations from the mean. That is, (m – 3*s, m + 3*s).

Step 2

Calculations:

The random variable x follows normal distribution with mean 50 and standard deviation 5.

That is, m = 50 and s = 5.

The calculation based on empirical rule is given below:

One \(\displaystyle\sigma\) limits:

\(\displaystyle{\left(\mu\pm\sigma\right)}={\left({5}--{5},{50}+{5}\right)}={\left({45},{55}\right)}\)

68% of scores are between 45 and 55.

Two sigma limits:

\(\displaystyle{\left(\mu\pm{2}\sigma\right)}={\left({50}-{2}\times{5},{50}+{2}\times{5}\right)}={\left({40},{60}\right)}\)

95% of scores are between 40 and 60.

Three \(\displaystyle\sigma\) limits:

\(\displaystyle{\left(\mu\pm{3}\sigma\right)}={\left({50}-{3}\times{5},{50}+{3}\times{5}\right)}={\left({35},{65}\right)}\)

99% of scores are between 35 and 65.

Step 3

Obtain the correct option:

From the given 5 options, option (B) states that 95% of the scores lie between 40 and 60.

Hence, option (B) is correct.

Step 4

Answer:

Option (B) is correct.

Empirical rule:

For symmetric or normal distributions,

68% of values fall within one standard deviation from the mean. That is, (m – s, m + s).

95% of values fall within two standard deviations from the mean. That is, (m – 2*s, m + 2*s).

7% of values fall within three standard deviations from the mean. That is, (m – 3*s, m + 3*s).

Step 2

Calculations:

The random variable x follows normal distribution with mean 50 and standard deviation 5.

That is, m = 50 and s = 5.

The calculation based on empirical rule is given below:

One \(\displaystyle\sigma\) limits:

\(\displaystyle{\left(\mu\pm\sigma\right)}={\left({5}--{5},{50}+{5}\right)}={\left({45},{55}\right)}\)

68% of scores are between 45 and 55.

Two sigma limits:

\(\displaystyle{\left(\mu\pm{2}\sigma\right)}={\left({50}-{2}\times{5},{50}+{2}\times{5}\right)}={\left({40},{60}\right)}\)

95% of scores are between 40 and 60.

Three \(\displaystyle\sigma\) limits:

\(\displaystyle{\left(\mu\pm{3}\sigma\right)}={\left({50}-{3}\times{5},{50}+{3}\times{5}\right)}={\left({35},{65}\right)}\)

99% of scores are between 35 and 65.

Step 3

Obtain the correct option:

From the given 5 options, option (B) states that 95% of the scores lie between 40 and 60.

Hence, option (B) is correct.

Step 4

Answer:

Option (B) is correct.