Stat1600

Solution to Midterm #1

Form D

1. The chest sizes, measured in inches, of Scottish militiamen in the early 19th century were recorded (adapted from DASL: Chest sizes of Militiamen). A histogram of chest sizes shows an approximately normal curve. It is known that the mean chest size is 39.4 inches with a standard deviation of 2.2 inches. Given this information, use the Z-Table provided to answer the following questions:
(a) (15 points) What percentage of soldiers having chest size under 37.2 inches?
Note that the z-score of 37.2 is
37.2 − 39.4
= −1.00
2.2

z=

Hence, the chance that a soldier having chest size under 37.2 inches can be computed as the area under the Z curve to the left of −1.00 which equals the area to the right of
1.00 and is calculated as 1−0.8413 =0.1587 (see figures below). That is, 15.87%.

=

= 1−0.8413
= 0.1587

=1−

−1.00

1.00

z

z

1.00 z (b) (10 points) What percentage of soldiers having chest size between 35 and 43.8 inches?
Note that the z-score for 35 is z= 35 − 39.4
= −2.00
2.2

and that the z-score for 43.8 is z= 43.8 − 39.4
= 2.00.
2.2

Hence the percentage of men soldiers having chest size between 35 and 43.8 inches is the area (in percentage) under the Z curve bounded by −2.00 and 2.00 which is approximately 95% according to the empirical rules.

1

Using Z Table, the probability is 0.9544 or 95.44%:

=1−
−2.00

2.00

=2x

−2.00

z

−1

2.00

= 2x0.9772 − 1
= 0.9544

2.00

z

z

(c) (10 points) A soldier had chest size measured at least as large as 60% of all Scottish militiamen. What was his chest size?
The soldier was at 60th or higher percentile. The 59.87th z-percentile is 0.25 (note: the area under Z curve to the left of 0.25 is 0.5987 and the area under Z curve to the left of 0.26 is 0.6026). Hence, the 60.26th