Question

the accompanying data represent the miles per gallon of a random sample of cars with a three - cylinder, 1.0 liter engine.
(a) compute the z - score corresponding to the individual who obtained 39.6 miles per gallon. interpret this result.
(b) determine the quartiles.
(c) compute and interpret the interquartile range, iqr.
(d) determine the lower and upper fences. are there any outliers?
click the icon to view the data.
(a) compute the z - score corresponding to the individual who obtained 39.6 miles per gallon. interpret this result.
the z - score corresponding to the individual is and indicates that the data value is standard deviation(s) the
(type integers or decimals rounded to two decimal places as needed.)
mpg data

Explanation:

Step1: Calculate the mean

First, sum all the data values: $32.3+36.9+37.5+38.6+39.9+42.6+34.4+36.3+36.1+38.9+40.6+42.4+34.8+37.4+38.2+39.2+41.5+43.6+36.8+37.7+38.5+39.6+41.6+48.9 = 879.8$. There are $n = 24$ data - points. The mean $\bar{x}=\frac{879.8}{24}\approx36.66$.

Step2: Calculate the standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(32.3 - 36.66)^{2}=(-4.36)^{2}=19.0096$, $(36.9 - 36.66)^{2}=(0.24)^{2}=0.0576$, etc.
Sum of squared differences $\sum_{i = 1}^{24}(x_{i}-36.66)^{2}=347.9784$.
$s=\sqrt{\frac{347.9784}{23}}\approx3.89$.

Step3: Compute the z - score

The formula for the z - score is $z=\frac{x-\bar{x}}{s}$. For $x = 39.6$, $z=\frac{39.6-36.66}{3.89}=\frac{2.94}{3.89}\approx0.75$.
The z - score of $0.75$ indicates that the data value is $0.75$ standard deviations above the mean.

Answer:

The z - score corresponding to the individual is $0.75$ and indicates that the data value is $0.75$ standard deviation(s) above the mean.