gpa\n3\n2.18\n3.66\n3.86\n3.31\n1.57\n3.52\n1.97\n3.79\n2.59\n3.15\n1.52\n2.59\n3.56\n2.23\n(a) find the…

Explanation:

Step1: Sort the data

$1.52, 1.57, 1.97, 2.18, 2.23, 2.59, 2.59, 3, 3.15, 3.31, 3.52, 3.56, 3.66, 3.79, 3.86$

Step2: Calculate the first quartile ($Q_1$)

There are $n = 15$ data - points. The position of $Q_1$ is $i=\frac{n + 1}{4}=\frac{15+1}{4}=4$. So $Q_1$ is the 4th value in the sorted data, $Q_1 = 2.18$.

Step3: Calculate the third quartile ($Q_3$)

The position of $Q_3$ is $i = 3\times\frac{n + 1}{4}=3\times\frac{15 + 1}{4}=12$. So $Q_3$ is the 12th value in the sorted data, $Q_3=3.56$.

Step4: Calculate the mean ($\bar{x}$)

$\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{1.52+1.57+1.97+2.18+2.23+2.59+2.59+3+3.15+3.31+3.52+3.56+3.66+3.79+3.86}{15}=\frac{42.4}{15}\approx2.83$.

Step5: Calculate the median

Since $n = 15$ (odd), the median is the $(\frac{n + 1}{2})$ - th value. $\frac{15+1}{2}=8$ - th value, so the median is $3$.

Step6: Calculate the range

Range = Maximum - Minimum. Maximum = $3.86$, Minimum = $1.52$. Range=$3.86 - 1.52 = 2.34$.

Step7: Calculate the sample standard deviation ($s$)

First, calculate the sum of squared differences from the mean. $\sum_{i = 1}^{n}(x_i-\bar{x})^2=(1.52 - 2.83)^2+(1.57 - 2.83)^2+\cdots+(3.86 - 2.83)^2$. $\sum_{i = 1}^{n}(x_i-\bar{x})^2=(-1.31)^2+(-1.26)^2+\cdots+(1.03)^2$. $\sum_{i = 1}^{n}(x_i-\bar{x})^2 = 1.7161+1.5876+\cdots+1.0609$. $\sum_{i = 1}^{n}(x_i-\bar{x})^2=13.978$. $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=\sqrt{\frac{13.978}{14}}\approx0.999\approx1.00$.

Step8: Calculate the inter - quartile range (IQR)

IQR=$Q_3 - Q_1=3.56 - 2.18 = 1.38$.

Step9: Calculate the lower and upper limits for outliers

Lower Limit (LL)=$Q_1-1.5\times IQR=2.18-1.5\times1.38=2.18 - 2.07 = 0.11$. Upper Limit (UL)=$Q_3+1.5\times IQR=3.56+1.5\times1.38=3.56 + 2.07 = 5.63$.

Step10: Calculate the sample variance ($s^2$)

$s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}=\frac{13.978}{14}\approx0.99986\approx1.00$.

Answer:

(a) $2.18$ (b) $3.56$ (c) $2.83$ (d) $3$ (e) $2.34$ (f) $1.00$ (g) $1.38$ (h) LL: $0.11$, UL: $5.63$ (i) $1.00$