Download Standard deviation as a ruler

Standard deviation as a ruler
When we are using the standard deviation as a ruler to measure how far an observation is above or below the
mean, we are using a STANDARDIZED SCORE, or z-score.
z=
x−x
s
Using standardized scores has many advantages. Since z-scores have no units, we can compare values that are
measured on different scales, with different units, or from different populations.
A negative Z-score tells you the data value is below the mean
A positive Z-score tells you the data is above the mean
Again a Z-score tells you “how many standard deviations a piece of data is from the mean that the piece
of data came from”
Examples:
1) Suppose that the distribution of male heights has a mean of 69 inches with a standard deviation of 3
inches and the distribution of female heights has a mean of 64 inches with a standard deviation of 2.5
inches.
a) Who is taller, relatively speaking, a 65 inch male or a 60 inch female?
b) An exclusive club only allows members who are especially tall. The height requirement for women is
72 inches. What is the equivalent requirement be for men?
2) A professional baseball team has the Money to purchase one additional player, they are in need of a
pitcher and a slugger (like most teams) through much research and negotiations they finally get the
decision down to two players:
The pitcher has a good track record and last season had 198 strikeouts
The slugger hit 38 homeruns last season
The average for the top 30 pitchers in the league last year was 190 strikeouts with a SD of 16
The average for the top 30 sluggers in the league last year was 32 homeruns with a SD of 4
WHO WOULD BE A BETTER USE OF THE TEAMS MONEY?
USING THE CALCULATOR AND THE NORMAL MODEL TO CALCULATE POSSIBILITIES
The command in the calculator that determines the percent of data between two z-scores in a normal
distribution is
Normalcdf(leftZ, rightZ)
it can be found by pressing 2nd VARS
Suppose that the distance a certain golfer can hit a golf ball is approximately normally distributed with a mean
of 250 yards and a standard deviation of 15 yards.
a) Sketch this distribution.
b) What proportion of his shots will go less than 231 yards?
c) What proportion will go at least 300 yards?
d) What proportion will go between 240 and 260 yards?
e) What is the 75th percentile for this distribution? *a percentile tells you what percent of data is at or
below a specific piece of data….for example if you took a test and was told you are in the 87th%ile, that
would mean that you did as good or better than 87% of all the people that also took the test .
f) What distance be exceeded 90% of the time by this golfer?