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 41.4 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?

32.7
34.0
34.7
35.4
36.0
36.2
37.3
37.6
37.7
37.9
38.1
38.5
38.6
39.0
39.2
39.4
39.9
40.7
41.4
41.8
42.5
42.8
43.7
49.0

Answer 1

Solution:given that 32.7,34.0,34.7,35.4,36.0,36.2,37.3,37.6,37.7,37.9,38.1,38.5,38.6,39.0,39.2,39.4,39.9,40.7,41.4,41.8,42.5,42.8,43.7,49.0

mean = 38.92, sd = 3.54

(a) Z-score : (41.4-38.92)/3.54 = 0.7006

(b) Quartiles :

First Quartiles : 36.75
Explanation

The first quartile (or lower quartile or 25th percentile) is the median of the bottom half of the numbers. So, to find the first quartile, we need to place the numbers in value order and find the bottom half.

32.7 34.0 34.7 35.4 36.0 36.2 37.3 37.6 37.7 37.9 38.1 38.5 38.6 39.0 39.2 39.4 39.9 40.7 41.4 41.8 42.5 42.8 43.7 49.0   

So, the bottom half is

32.7 34.0 34.7 35.4 36.0 36.2 37.3 37.6 37.7 37.9 38.1 38.5   

The median of these numbers is 36.75

Second Quartiles : 38.55

Explanation

The median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

32.7 34.0 34.7 35.4 36.0 36.2 37.3 37.6 37.7 37.9 38.1 38.5 38.6 39.0 39.2 39.4 39.9 40.7 41.4 41.8 42.5 42.8 43.7 49.0   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

median = (38.5+38.6)/2 = 38.55

Third quartile : 41.05

Explanation

The third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers. So, to find the third quartile, we need to place the numbers in value order and find the upper half.

32.7 34.0 34.7 35.4 36.0 36.2 37.3 37.6 37.7 37.9 38.1 38.5 38.6 39.0 39.2 39.4 39.9 40.7 41.4 41.8 42.5 42.8 43.7 49.0   

So, the upper half is

38.6 39.0 39.2 39.4 39.9 40.7 41.4 41.8 42.5 42.8 43.7 49.0   

The median of these numbers is 41.05.

(c)

interpret the interquartile range, IQR : 4.3

Explanation

The interquartile range is the difference between the third and first quartiles.

The third quartile is 41.05.

The first quartile is 36.75.

The interquartile range = 41.05 - 36.75 = 4.3.

(d)

Lower fence: 30.3 , Upper fence: 47.5

Outliers : 49