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 suni 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: \(75 + 80+90 + 100+75 + 85+80 + 80+90 + 85=820\).
There are \(n = 10\) scores.
The mean \(\bar{x}=\frac{820}{10}=82\).

Step2: Calculate the standard deviation

The formula for the standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}\).
\((75 - 82)^{2}=49\), \((80 - 82)^{2}=4\), \((90 - 82)^{2}=64\), \((100 - 82)^{2}=324\), \((75 - 82)^{2}=49\), \((85 - 82)^{2}=9\), \((80 - 82)^{2}=4\), \((80 - 82)^{2}=4\), \((90 - 82)^{2}=64\), \((85 - 82)^{2}=9\).
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=49+4 + 64+324+49+9+4+4+64+9 = 640\).
\(s=\sqrt{\frac{640}{10}}=\sqrt{64}=8\).

Step3: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\bar{x}}{s}\).
For \(x = 100\), \(\bar{x}=82\), \(s = 8\).
\(z=\frac{100 - 82}{8}=\frac{18}{8}=2.2\).

Answer:

d. 2.2