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 68-95-99.7 rule is only used for problems 1 and 2.

1. The mean is 100 points, standard deviation is 10 points.
   1. 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.
   2. 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.
2. The mean is 6 inches, standard deviation is 1.5 inches.
   1. Find the proportion of values below 6 inches. Since 6 is the mean, the proportion below it is .5.
   2. 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.
3. The mean is 20 cigarettes, standard deviation is 5 cigarettes.
   1. Find the proportion of values above 22.5 cigarettes. z=(22.5-20)/5 =.5. The proportion above +.5 = proportion below -.5 = .3085.
   2. 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.
4. The mean is 165 lbs., standard deviation is 12 lbs.
   1. Find the proportion of values below 148 lbs. z-value is (148-165)/12 = -1.42. The proportion of values below z=-1.42 is .0778.
   2. 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.
5. The mean is 4 people, standard deviation is 1.3 people. Find the proportion of values less than 2 people. z = (2-4)/1.3 = -1.54; the proportion below is .0618.
6. The mean is 11 years, standard deviation is 2 years.
   1. Find the proportion of values less than 17 years. z = (17-11)/2 = 3; the proportion below +3 is .9987.
   2. 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.
7. The mean is $30,000; standard deviation is $8,000. Find the proportion of values between $20,000 and $22,000. The z-values are -1.25 and -1.00, corresponding to proportions .1056 and .1587. Subtracting, we get .0531.
8. 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.
9. 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.
10. The mean is 60 degrees, standard deviation is 16 degrees. Find the proportion of values below 79 degrees. The z-value is +1.19. The proportion above +1.19 equals the proportion below -1.19, which is .8830.
11. The mean is 300 ml., standard deviation is 3 ml. Find the proportion of values
    1. less than 280 ml. This is almost 7 standard deviations below; practically none (proportion approx. 0) are below this.
    2. greater than 280 ml. Practically all (proportion approx. 1) are above.
    3. less than 315 ml. This is 5 standard deviations above; practically all (proportion approx. 1) are below this
    4. greater than 315 ml. Practically none (proportion approx. 0) are above this.

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