Question

bell ringer - sep 22

1. find the z - scores of each score in the following data.

23 56 70 45 29 38 71 40
80 55

1. find the first and third quartiles of the following data.

98 45 67 55 30 88 70 48
76 80

Explanation:

Step1: Calculate the mean of the first data - set

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
For the data - set $23,56,70,45,29,38,71,40,80,55$, $n = 10$ and $\sum_{i=1}^{10}x_{i}=23 + 56+70+45+29+38+71+40+80+55 = 507$.
So, $\bar{x}=\frac{507}{10}=50.7$.

Step2: Calculate the standard deviation of the first data - set

The formula for the standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(23 - 50.7)^{2}=(-27.7)^{2}=767.29$, $(56 - 50.7)^{2}=(5.3)^{2}=28.09$, $(70 - 50.7)^{2}=(19.3)^{2}=372.49$, $(45 - 50.7)^{2}=(-5.7)^{2}=32.49$, $(29 - 50.7)^{2}=(-21.7)^{2}=470.89$, $(38 - 50.7)^{2}=(-12.7)^{2}=161.29$, $(71 - 50.7)^{2}=(20.3)^{2}=412.09$, $(40 - 50.7)^{2}=(-10.7)^{2}=114.49$, $(80 - 50.7)^{2}=(29.3)^{2}=858.49$, $(55 - 50.7)^{2}=(4.3)^{2}=18.49$.
$\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=767.29+28.09+372.49+32.49+470.89+161.29+412.09+114.49+858.49+18.49 = 3236.1$.
$s=\sqrt{\frac{3236.1}{9}}\approx\sqrt{359.567}\approx18.96$.

Step3: Calculate the z - scores of the first data - set

The formula for the z - score is $z=\frac{x-\bar{x}}{s}$.
For $x = 23$, $z=\frac{23 - 50.7}{18.96}\approx - 1.46$.
For $x = 56$, $z=\frac{56 - 50.7}{18.96}\approx0.28$.
For $x = 70$, $z=\frac{70 - 50.7}{18.96}\approx1.02$.
For $x = 45$, $z=\frac{45 - 50.7}{18.96}\approx - 0.30$.
For $x = 29$, $z=\frac{29 - 50.7}{18.96}\approx - 1.14$.
For $x = 38$, $z=\frac{38 - 50.7}{18.96}\approx - 0.67$.
For $x = 71$, $z=\frac{71 - 50.7}{18.96}\approx1.07$.
For $x = 40$, $z=\frac{40 - 50.7}{18.96}\approx - 0.56$.
For $x = 80$, $z=\frac{80 - 50.7}{18.96}\approx1.55$.
For $x = 55$, $z=\frac{55 - 50.7}{18.96}\approx0.23$.

Step4: Sort the second data - set

Sort the data - set $98,45,67,55,30,88,70,48,76,80$ in ascending order: $30,45,48,55,67,70,76,80,88,98$.

Step5: Calculate the position of the first and third quartiles

The formula for the position of the $p$ - th percentile is $L=(n + 1)\frac{p}{100}$, where $n$ is the number of data points.
For the first quartile ($p = 25$) and $n = 10$, $L=(10 + 1)\frac{25}{100}=2.75$.
The first quartile $Q_{1}$ is the value between the 2nd and 3rd ordered data points. $Q_{1}=45+(48 - 45)\times0.75 = 45 + 2.25=47.25$.
For the third quartile ($p = 75$), $L=(10 + 1)\frac{75}{100}=8.25$.
The third quartile $Q_{3}$ is the value between the 8th and 9th ordered data points. $Q_{3}=80+(88 - 80)\times0.25=80 + 2=82$.

Answer:

1. The z - scores are approximately: - 1.46, 0.28, 1.02, - 0.30, - 1.14, - 0.67, 1.07, - 0.56, 1.55, 0.23.
2. The first quartile $Q_{1}=47.25$ and the third quartile $Q_{3}=82$.