Question

the table below shows the scores on a science test.
john 75 brett 80
elizabeth 90 cindy 100
jake 75 kenji 85
rick 80 joy 80
raul 90 sunl 85
the number of standard deviations between a score and the mean score is indicated by a z - score. what was cindys z - score on the science test?
a - 2.2 c. 16
b. 100 d. 2.2

Explanation:

Step1: Calculate the mean

First, find the sum of all scores: \(75 + 80+90 + 100+75 + 85+80 + 80+90 + 85=840\).
There are \(n = 10\) scores. The mean \(\bar{x}=\frac{840}{10}=84\).

Step2: Calculate the standard deviation

The formula for the standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}\).
\((75 - 84)^{2}=81\), \((80 - 84)^{2}=16\), \((90 - 84)^{2}=36\), \((100 - 84)^{2}=256\), \((75 - 84)^{2}=81\), \((85 - 84)^{2}=1\), \((80 - 84)^{2}=16\), \((80 - 84)^{2}=16\), \((90 - 84)^{2}=36\), \((85 - 84)^{2}=1\).
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=81+16 + 36+256+81+1+16+16+36+1 = 540\).
\(s=\sqrt{\frac{540}{10}}=\sqrt{54}\approx7.35\).

Step3: Calculate the z - score

The z - score formula is \(z=\frac{x-\bar{x}}{s}\).
For \(x = 100\), \(\bar{x}=84\), \(s\approx7.35\).
\(z=\frac{100 - 84}{7.35}=\frac{16}{7.35}\approx2.2\).

Answer:

d. 2.2