Name _____________________________ Period ____
Alg 2
Normal Distribution Practice
1. Answer the following questions for a normal distribution.
a. What percent of values lie below the mean? 50%
b. What percent of values lie within one standard deviation of the mean? 68%
c. What percent of values lie within two standard deviations of the mean? 95%
d. What percent of values lie within three standard deviations of the mean? 99.7%
e. What percent of values are more than 2 standard deviations greater than the mean?
2.5% if subtracting 95% (percent within 2σ) from 100% then dividing by 2
2.28% if using the z-table
f. What percent of values are less than 1 standard deviation below the mean?
15.87%  this value is from the z-table
For # 2 – 3 sketch and label the normal curve for the following data. Make a prediction based on
the curve using the 68-95-99.7 Rule.
2. A light bulb lasts an average of 219 h. Out of 1000 bulbs, how many will not last 79 h if the standard
deviation is 70 h?
If you use the curve to approximate: Your probability is 2.5%. So 2.5% of 1000 is 25 bulbs.
If you use the z-table to calculate: Your probability is 2.28%. So 2.28% of 1000 is 22.8 bulbs.
3. In a math class of 26 students, a series of 100 multiplication problems can be completed in a mean time
of 4 min. The standard deviation is 1 min. How many math students will still be working after 5 min?
Students at 5min. are at 1σ above the mean. A z-score of 1 has a probability of 84.13% so
100-84.13 = 15.87% (which is the probability students will be working after 5 min.).
So, 15.87% of 26 is 4.13. Approximately 4 students are still working after 5 minutes.
Find the following probabilities using the Z-score table.
4. P  z  3.15
6. P  z  1.77 
7. P  0.48  z  2.97 
Probability below 3.15
Find 3.15 on z-table
Probability above 1.77
Find 1.77 on z-table  .9616
Then subtract from 1.
Probability between -0.48 & 2.97
Find 2.97 on z-table  .9985
Find -0.48 on z-table  .3156
Subtract .9985 - .3156
99.92%
3.84%
68.29%
5. 500 juniors at Central High School took the ACT last year. The scores were distributed normally
with a mean of 24 and a standard deviation of 4. Label the mean and three standard deviations from
the mean. Use 68-95-99.7 Rule for #a – h.
a. What percentage of scores are between 20 and
28? 68% b/c 20 is at -1σ and 28 is at 1 σ
b. What percentage of scores are between 16 and
32? 95% b/c 16 is at -2σ and 32 is at 2σ
c. What percentage of scores are between 16 and
28?
81.5% b/c 16 is at -2σ and 28 is at 1σ
d. What percentage of scores are less than 12?
0.13% b/c 12 is at -3σ
e. What percentage of scores are less than 24?
50% b/c 24 is the mean
f. Approximately how many juniors scored between 24 and 28?
34% between 24 and 28 b/c 24 is the mean and 28 is at 1σ, so 34% of 500 is 170
g. Approximately how many juniors scored between 20 and 28?
68% between 20 and 28 b/c 20 is at -1σ and 28 is at 1σ, so 68% of 500 is 340
h. Approximately how many juniors scored between 16 and 20?
13.5% between 16 and 20 b/c 16 is at -2σ and 20 is at -1σ, so 13.5% of 500 is 67.5
You will need to use your z-score table for the rest of this problem
i. What percentage of scores are less than 30?
z-score of 30 is 1.5, so 1.5 in z-table is 93.32%
j. What percentage of scores are greater than 18?
z-score of 18 is -1.5, so -1.5 in z-table is 0.068. Subtract from 1 to get 93.32%.
k. How many students scored above a 27?
z-score of 27 is 0.75, so 0.75 in z-table is .7734. Subtract from 1 to get .2266. 22.66%
of 500 students is 113.3 students.
l.
How many students scored below 22?
z-score of 22 is -0.5, so -0.5 in z-table is .3085. 30.85% of 500 students is 154.25 students.
m. What percentage scored between 13 and 29?
z-score of 13 is -2.75, so -2.75 in z-table is .0030. z-score of 29 is 1.25. So 1.25 in z-table is
.8944. So .8944 - .0030 = 0.8914 or 89.14%
n. What percentage scored between 24 and 33?
z-score of 24 is 0, so 0 in z-table is .5000. z-score of 33 is 2.25. So 2.25 in z-table is .9878.
So .9878 - .5 = 0.4878 or 48.78%