Question

a) TRUE OR FALSE: For a set of data with a mean of 18 and a variance of 25, approximately 68% of the values will fall between 13 to 23.

b) Which of the following statement is true about the relationship between a sample and a population

1)Every sample is a perfect representation of a population

2)The sample size is smaller than a population size

3)Every member of a population is also in the sample

4)A population size is smaller than a sample size

c) If a data set showing types of pizza ordered at a particular restaurant indicates 24 out of 72 orders were for pepperoni pizza, what percentage would represent pepperoni pizza in a pie chart?

d) In constructing a Pareto chart for the following distribution which category comes first:

Year in School	Number of Students
Freshmen	28
Sophomores	14
Juniors	40
Seniors	18

e) If a frequency distribution had class limit of 31 –47, what would be the class width?

f) According to Chebyshev's theorem, the minimum proportion of data values from a data set that are within 4 standard deviations from the mean is _________.

g) If a data set has 25 values and a standard deviation 8.4, then the variance is ________.

h) Identify the five-number summary of the following data set.
7, 11, 21, 28, 32, 33, 37, 43.

I) Find the sample standard deviation of the following set of values.
12, 34, 13, 17, 23, 31, 13

Answer 1

a) TRUE OR FALSE: For a set of data with a mean of 18 and a variance of 25, approximately 68% of the values will fall between 13 to 23.

In 68% +/- sd observation lies

( 18 - 5 , 18+ 5 ) = (13,23)

TRUE

b) Which of the following statement is true about the relationship between a sample and a population

2)The sample size is smaller than a population size

c) If a data set showing types of pizza ordered at a particular restaurant indicates 24 out of 72 orders were for pepperoni pizza, what percentage would represent pepperoni pizza in a pie chart?

% pepperoni pizza = (24/72)*100 = 33.33%

Angle in pie chart = % pepperoni pizza * 360 = 0.33*360 = 120

d)

Now, the following table shows the categories in descending order (with respect to the frequencies), along with the cumulative relative frequencies

Categories	Frequencies	Cum. Relative Frequencies (%)
Juniors	40	40
Freshmen	28	68
Seniors	18	86
Sophomores	14	100
Total =	100

Therefore, the following Pareto Chart is obtained based on the table above:

junior comes first

e) If a frequency distribution had class limit of 31 –47, what would be the class width?

Class width = 47 - 31 = 16

f) According to Chebyshev's theorem, the minimum proportion of data values from a data set that are within 4 standard deviations from the mean is _________.

k = 4

g) If a data set has 25 values and a standard deviation 8.4, then the variance is ________

variance = standard deviation^2 = 8.4^2

variance = 70.56

h) Identify the five-number summary of the following data set.
7, 11, 21, 28, 32, 33, 37, 43

These are the sample data that have been provided with:

Position	X (Asc. Order)
1	7
2	11
3	21
4	28
5	32
6	33
7	37
8	43

Based on the table above, the minimum is min=7 and the maximum is max=43. Now the position of the first quartile Q_1 is:

  

Since L25​=2.25 is not an integer number, the first quartile Q_1 is computed by interpolating between the values located in the 2nd and 3rd positions, as shown in the formula below:

Since the sample size n = 8 is even, we have that (n+1)/2 = (8+1)/2 = 4.5 is not an integer value, so the median is computed directly by finding the average of the values located at positions 4th and 5th, which is:

  

Now the position of the third quartile Q_3 ​ is:

  

Since L75​=6.75 is not an integer number, the third quartile Q3​ is computed by interpolating between the values located in the 6th and 7th positions, as shown in the formula below:

  

Therefore, based on the results found above, we get the following five-number summary:

Minimum =	7
Q1​ =	22.75
Median =	32.5
Q3​ =	41.5
Maximum =	43

I) Find the sample standard deviation of the following set of values.
12, 34, 13, 17, 23, 31, 13

The sample size is n = 7. The provided sample data along with the data required to compute the sample variance s2 are shown in the table below:

	X	X2
	12	144
	34	1156
	13	169
	17	289
	23	529
	31	961
	13	169
Sum =	143	3417

The sample variance s2 is

Therefore, the sample standard deviation s is