Question

a standardized exams scores are normally distributed. in a recent year, the mean test score was 20.7 and the standard deviation was 5.7. the test scores of four students selected at random are 12, 21, 8, and 36. find the z - scores that correspond to each value and determine whether any of the values are unusual. (round to two decimal places as needed.) the z - score for 21 is 0.05. (round to two decimal places as needed.) the z - score for 8 is - 2.23. (round to two decimal places as needed.) the z - score for 36 is 2.68. (round to two decimal places as needed.) which values, if any, are unusual? select the correct choice below and, if necessary, fill in the answer box within your choice. a. the unusual value(s) is/are. (use a comma to separate answers as needed.) b. none of the values are unusual.

Explanation:

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data - point, $\mu$ is the mean, and $\sigma$ is the standard deviation. Here, $\mu = 20.7$ and $\sigma=5.7$.

Step2: Determine unusual values criteria

Unusual values in a normal distribution are typically considered to be those with a z - score less than - 2 or greater than 2.

Step3: Analyze z - scores

The z - score for $x = 12$ is $z_1=\frac{12 - 20.7}{5.7}=\frac{-8.7}{5.7}\approx - 1.53$.
The z - score for $x = 21$ is $z_2=\frac{21 - 20.7}{5.7}=\frac{0.3}{5.7}\approx0.05$.
The z - score for $x = 8$ is $z_3=\frac{8 - 20.7}{5.7}=\frac{-12.7}{5.7}\approx - 2.23$.
The z - score for $x = 36$ is $z_4=\frac{36 - 20.7}{5.7}=\frac{15.3}{5.7}\approx2.68$.
Since $z_3\approx - 2.23\lt - 2$ and $z_4\approx2.68\gt2$, the values 8 and 36 are unusual.

Answer:

A. 8, 36