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 38.4 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? 32.5; 35.9; 37.6; 38.6; 40.4; 42.5; 34.0; 36.2; 37.8; 38.9; 40.6; 42.6; 34.7; 37.3; 38.1; 39.4 ;41.3; 43.4; 35.6; 37.4; 38.4; 39.7; 41.8; 49.1

Answer 1

(a)

From the given data, the following statistics are calculated:

n = 24

= 38.9083

s = 3.5485

For X = 38.4:

Z =(38.4 - 38.9083)/3.5485

= - 0.1432

Table of Area Under Standard Normal Curve gives area = 0.0557

So,

P(X<38.4) = 0.5 - 0.0557 = 0.4443=44.43 %

So,

Answer is:
(i) Z = - 0.1432

(ii) 44.43% of values lie below X = 38.4

(b)

Arranging numbers in ascending order, we get:

32.5, 34.0, 34.7, 35.6, 35.9 36.2 37.3, 37.4, 37.6, 37.8, 38.1, 38.4, 38.6, 38.9, 39.4, 39.7, 40.4, 40.6 41.3, 41.8, 42.5,42.6, 43.4, 49.1

Bottom half is given by:

32.5, 34.0, 34.7, 35.6, 35.9 36.2 37.3, 37.4, 37.6, 37.8, 38.1, 38.4

Middle of these numbers is:

36.75

So,

First Quartile = Q1 = 36.75

n = 24

Median is (24 + 1)/2th item = Average of 12th & 13th items = (38.4 + 38.6)/2 = 38.5

So,

Second Quartile = Q2 = Median = 38.5

Upper Half is given by:

8.6, 38.9, 39.4, 39.7, 40.4, 40.6 41.3, 41.8, 42.5,42.6, 43.4, 49.1

Middle of these numbers is:

40.95

So,

Third Quartile = Q3 = 40.95

So,

Answers are:

First Quartile = Q1 = 36.75

Second Quartile = Q2 = Median = 38.5

Third Quartile = Q3 = 40.95

(c)

Interquartile Range = IQR = Q3-Q1 = 40.95 - 36.75 = 4.2

Interquartile Range= 4.2 is the range of the middle 50% of the data.

So,

Interquartile Range = 4.2

Interquartile Range= 4.2 is the range of the middle 50% of the data.

(d)

Lower fence=Q1-1.5 IQR = 36.75 - (1.5 X 4.2) = 36.75 - 6.3 = 30.45

Upper fence=Q3 + 1.5 IQR = 40.95 + (1.5 X 4.2) = 40.95 + 6.3 = 47.25

Since 49.1 is greater than Upper fence = 47.25, the value of 49.1 is an outlier.

So,

Answers are:

Lower fence=Q1-1.5 IQR = 30.45

Upper fence=Q3 + 1.5 IQR = 47.25

49.1 is an outlier.