
Stat 200
Normal Exercise Solutions
Assume all the data sets in these exercises follow a normal curve closely enough that a
normal approximation may be used. Use sketches as much as possible. Note:
the 689599.7 rule is only used for problems 1 and 2.
 The mean is 100 points, standard deviation is 10 points.
 Find the proportion of values below 70 points. 70 is 3 standard
deviations below. Since 99.7% are within 3 sd's, that leaves
half of 100% minus 99.7% below: .3%/2=.15%, or .0015.
 The middle 95% of the values are between what two point values? The
middle 95% are within 2 standard deviations of the mean: between 80 and 120.
 The mean is 6 inches, standard deviation is 1.5 inches.
 Find the proportion of values below 6 inches. Since 6 is the mean, the
proportion below it is .5.
 The shortest 16% are shorter than how many inches? Since the middle 68%
are between 4.5 and 7.5, the shortest 16% are below 4.5.
 The mean is 20 cigarettes, standard deviation is 5 cigarettes.
 Find the proportion of values above 22.5 cigarettes. z=(22.520)/5
=.5. The proportion above +.5 = proportion below .5 = .3085.
 The top 10% are greater than how many cigarettes? 10% above means 90%, or
.9, are below. .9000 (actually .8997) corresponds to z=+1.28. Unstandardized,
we have 20+1.28(5) = 26.4.
 The mean is 165 lbs., standard deviation is 12 lbs.
 Find the proportion of values below 148 lbs. zvalue is
(148165)/12 = 1.42. The proportion of values
below z=1.42 is .0778.
 The lightest 2% are lighter than how many pounds? .0200 (actually .0202)
below corresponds to z= 2.05. The unstandardized value is 165  2.05(12) =
140.4.
 The mean is 4 people, standard deviation is 1.3 people. Find the proportion of values
less than 2 people. z = (24)/1.3 = 1.54; the proportion below is .0618.
 The mean is 11 years, standard deviation is 2 years.
 Find the proportion of values less than 17 years. z = (1711)/2 = 3; the
proportion below +3 is .9987.
 The longest 7% are longer than how many years? .07 above means .93 are
below. The closest table value is .9306, corresponding to z = +1.48. The
unstandardized value is 11 + 1.48(2) = 13.96.
 The mean is $30,000; standard deviation is $8,000. Find the proportion of values between
$20,000 and $22,000. The zvalues are 1.25 and 1.00, corresponding to
proportions .1056 and .1587. Subtracting, we get .0531.
 The mean is 120 mm., standard deviation is 20 mm. Find the proportion of values greater than
112 mm. z is .4. The proportion above .4 equals the proportion below +.4,
which is .6554.
 The mean is 6 feet, standard deviation is .2 feet. Find the proportion of values greater
than 6.5 feet. z is 2.5. The proportion above +2.5 equals the proportion below
2.5, which is .0062.
 The mean is 60 degrees, standard deviation is 16 degrees. Find the proportion of values below
79 degrees. The zvalue is +1.19. The proportion above +1.19 equals the
proportion below 1.19, which is .8830.
 The mean is 300 ml., standard deviation is 3 ml. Find the proportion of values
 less than 280 ml. This is almost 7 standard deviations below; practically
none (proportion approx. 0) are below this.
 greater than 280 ml. Practically all (proportion approx. 1) are above.
 less than 315 ml. This is 5 standard deviations above; practically all
(proportion approx. 1) are below this
 greater than 315 ml. Practically none (proportion approx. 0) are above
this.
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