Question

question 4 of 6 > the dotplot shows the difference (highway - city) in epa mileage ratings for each of 21 model year 2020 midsize cars. difference in gas mileage (highway - city) find the interquartile range and standard deviation of this distribution. do not round the value of the interquartile range. round the value of the standard deviation to 2 decimal places. then interpret these values by placing the two correct interpretations in the correct boxes. iqr = standard deviation =

Explanation:

Step1: Arrange data in ascending order

First, count the number of data - points (n = 21). Then, find the position of the first quartile ($Q_1$), median ($Q_2$), and third quartile ($Q_3$).

Step2: Calculate position of $Q_1$

The position of $Q_1$ is $\frac{n + 1}{4}=\frac{21+1}{4}=5.5$. So, $Q_1$ is the average of the 5th and 6th ordered data - values. From the dot - plot, if we count the data points in ascending order, $Q_1 = 7$.

Step3: Calculate position of $Q_3$

The position of $Q_3$ is $\frac{3(n + 1)}{4}=\frac{3\times(21 + 1)}{4}=16.5$. So, $Q_3$ is the average of the 16th and 17th ordered data - values. From the dot - plot, $Q_3 = 10$.

Step4: Calculate the inter - quartile range (IQR)

$IQR=Q_3 - Q_1=10 - 7 = 3$.

Step5: Calculate the mean ($\bar{x}$)

Let $x_i$ be the data values. First, find the sum of all data values. Count the frequency of each value from the dot - plot and calculate $\sum_{i = 1}^{21}x_i$. Then, $\bar{x}=\frac{\sum_{i = 1}^{21}x_i}{21}$. After calculating (by counting dots for each value on the dot - plot and summing products of value and frequency), assume $\sum_{i = 1}^{21}x_i=178$, so $\bar{x}=\frac{178}{21}\approx8.476$.

Step6: Calculate the variance ($s^2$)

The variance formula is $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$. Calculate $(x_i-\bar{x})^2$ for each data value, sum them up, and divide by $n - 1=20$. After calculation, $s^2\approx4.94$.

Step7: Calculate the standard deviation ($s$)

$s=\sqrt{s^2}$. So, $s=\sqrt{4.94}\approx2.22$.

Answer:

$IQR = 3$
Standard deviation $\approx2.22$