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1. When a distribution is bell-shaped, approximately what percentage of data values will fall within one
standard deviation of the mean?
A. 50%
B
95%
C. 68%
D. 99.7%
2. What is the relationship between the variance and the standard deviation? What are the symbols used
to represent the sample variance and standard deviation? Why is the unbiased estimator of variance used?
Answer:
Standard deviation is the square root of variance.
S2 is the sample variance and s is the sample standard deviation.
When the population variance is unknown, sample variance is used as the unbiased estimator of
population variance
3. Nine households had the following number of children per household:
2, 0, 2, 2, 1, 2, 4, 3, 2
Find the mean, median, mode, and midrange for these data.
Solution:
Arrange data in increasing order: 0, 1, 2,2,2,2,2,3,4
Mean 
0 1 2  2  2  2  2  3 4
2
9
Mode = 2
Median = 2
Midrange = (0+4)/2 = 2
4. a) Use Chebyshev's theorem to find what percent of the values will fall between 120 and 150 for a data
set with mean of 135 and standard deviation of 7.5.
Solution:
According to Chebyshev’s theorem:
At least 75% of the data fall in the interval from  - 2 and  + 2
Since the standard deviation is 7.5, then 2 = 15
Since the mean is 135, then  - 2 = 135 – 15 = 120, and  + 2 = 135 + 15 = 150
From the above we can see that at least 75% of the data fall in the interval between 120 and
150
b) Use the Empirical Rule to find what two values 95% of the data will fall between for a data
set with mean 234 and standard deviation of 12.
Solution: The Empirical Rule states that “approximately 95.4% of the data values will lie within
2 standard deviations on each side of the mean”; therefore the values we are looking for lie
between  - 2 and  + 2
 - 2 = 234 – 2(12) = 234 – 24 = 210
 + 2 = 234 + 2(12) = 234 + 24 = 258
5. Find P25 for the following data:
2 6 3 4 2 1 2 0 1 3
Solution:
6
3
25th Percentile = Q1, therefore to find the 25th Percentile, we re-arrange the values in
increasing order
0
1
1
2
2
2
3
3
3
4
6
6
We have N(number of data values)=12 (ie even number of values)
So Median is the average of the 2 middle values = (2 +3) / 2 = 2.5.
This is Q2 (Second Quartile or the 50th Percentile)
The First Quartile (Q1 or the 25th Percentile) is defined as: the median of the lower half of the
data), so it is the median of the following set of values:
0
1
1
2
2
Since we have 5 data values (Odd number), then the median is the middle value which is: 1
Therefore, the 25th Percentile = 1
6. You are given the following frequency distribution for the number of errors 10 students made in a test
Errors
Frequency
0-2
14
3-5
13
6-8
11
9-11
8
12-14
9
Find: a. mean
b. variance
Solution:
Errors
0-2
3-5
6-8
9-11
12-14
Total
c. standard deviation.
Midpoint
1
4
7
10
13
35
Frequency
14
13
11
8
9
55
Midpoint * frequency
14
52
77
80
117
340
Mean = 340/55 = 6.18
Variance = [14(1-6.18)2+13(4-6.18)2+11(7-6.18)2+8(10-6.18)2+9(13-6.18)2] / 55 = 17.82
Standard deviation =
17.82  4.22
7. An aptitude test has a mean of 220 and standard deviation of 10. Find the corresponding z score for:
a) a test score of 232
b) a test score of 212
Solution:
Z score =
x 

a) a test score of 232
=
232  220 12

10
10
= 1.2
b) a test score of 212
=
212  220
8

10
10
= -0.8
8. You are given the following data.
x
y
2
14
3
13
4
11
5
8
5
9
7
4
7
3
Find
a.
SS(x)
b.
SS(y)
c.
SS (xy)
d.
The linear correlation coefficient, r
e.
The slope b1
f.
The y-intercept, b0
g.
The equation of the line of best fit.
Solution:
a) SS(x) = Σx2-(Σ x) 2/n = 177-332/7 = 21.429
b) SS(y) = Σy2-(Σ y) 2/n = 656-622/7 = 106.857
c) SS (xy) = Σ xy-(Σ x) (Σ y) /n = 245-(33)(62)/7 = -47.286
d) r = SS(x y) / √(SS(x)SS(y)) = -47.286 / √(21.429)(106.857) = -0.988
e) The slope b1 = SS (xy) / SS(x) = -47.286 / 21.429 = -2.207
f) The y-intercept, b0 = ybar - bxbar = 8.857 - (-2.207) (4.714) = 19.26
g) The equation of the line of best fit is y = 19.26 - 2.207x