Question

on a nationwide test taken by high school students, the mean score was 52 and the standard deviation was 13. the scores were normally distributed. complete the following statements.
(a) approximately 68% of the students scored between ■ and ■.
(b) approximately ■ of the students scored between 26 and 78.

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\sigma\)) of the mean (\(\mu\)), i.e., \(\mu-\sigma\) and \(\mu + \sigma\)
• Approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean, i.e., \(\mu - 2\sigma\) and \(\mu+2\sigma\)
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations of the mean, i.e., \(\mu- 3\sigma\) and \(\mu + 3\sigma\)

Given \(\mu = 52\) and \(\sigma=13\)

Step2: Solve part (a)

For \(68\%\) of the data (within \(1\) standard deviation):
Lower bound: \(\mu-\sigma=52 - 13=39\)
Upper bound: \(\mu+\sigma=52 + 13=65\)

Step3: Solve part (b)

Check the number of standard deviations \(26\) and \(78\) are from the mean.
For \(x = 26\): \(z=\frac{26 - 52}{13}=\frac{- 26}{13}=-2\)
For \(x = 78\): \(z=\frac{78 - 52}{13}=\frac{26}{13}=2\)
Since \(26=\mu - 2\sigma\) and \(78=\mu+2\sigma\), by the empirical rule, approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean.

Answer:

(a) \(39\) and \(65\)
(b) \(95\%\)