SOLUTION: The following scores represent the final examination grades for an elementary statistics course: 23 60 79 32 57 74 52 70 82 36 80 77 81 95 41 65 92 85 55 76 52 10 64 75 78

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Question 1202369: The following scores represent the final examination grades for an elementary
statistics course:
23 60 79 32 57 74 52 70 82 36
80 77 81 95 41 65 92 85 55 76
52 10 64 75 78 25 80 98 81 67
41 71 83 54 64 72 88 62 74 43
60 78 89 76 84 48 84 90 15 79
34 67 17 82 69 74 63 80 85 61
a) Make a stem-and-leaf display for these data. Show at least five stems. Sort
the data on each stem in order of increasing magnitude.
b) Find the median, lower quartile, and upper quartile of these data.
c) What fraction of the class received scores which were less than 65?
d) Make a frequency table, starting the first class interval at a lower class
boundary of 9.5. Use Sturges’ Rule.
e) Draw a frequency histogram.
f) Draw a relative frequency histogram on the same x-axis.
g) Draw a cumulative frequency diagram.
h) Find the mode.
i) Show a box plot of these data.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

I'll do the first three parts to get you started.

Part (a)

Given data set

23 60 79 32 57 74 52 70 82 36
80 77 81 95 41 65 92 85 55 76
52 10 64 75 78 25 80 98 81 67
41 71 83 54 64 72 88 62 74 43
60 78 89 76 84 48 84 90 15 79
34 67 17 82 69 74 63 80 85 61

There are 6 rows and 10 columns to give 6*10 = 60 values total.

Let's sort the data from smallest to largest. Use a spreadsheet to do this quickly.

10 15 17 23 25 32 34 36 41 41
43 48 52 52 54 55 57 60 60 61
62 63 64 64 65 67 67 69 70 71
72 74 74 74 75 76 76 77 78 78
79 79 80 80 80 81 81 82 82 83
84 84 85 85 88 89 90 92 95 98

From here, let's break up the data values so that each line has the same tens place value.

10,15,17,
23,25,
32,34,36,
41,41,43,48
52,52,54,55,57,
60,60,61,62,63,64,64,65,67,67,69,
70,71,72,74,74,74,75,76,76,77,78,78,79,79
80,80,80,81,81,82,82,83,84,84,85,85,88,89,
90,92,95,98
Then each stem is a different tens value. The stems are: 1,2,3,4,5,6,7,8,9
They represent 10,20,30,40,50,60,70,80,90 respectively.

Stem Leaf
1
2
3
4
5
6
7
8
9

To fill out the remaining entries, we will write the units digit of each value.
Examples:
The value 10 has stem 1 and leaf 0
The value 15 has stem 1 and leaf 5
The value 17 has stem 1 and leaf 7

I'll separate each leaf by a comma. Some textbooks will use a space as the separater. Other times there isn't a separater at all (and all of the leaves on one line are crammed together as one long string of numbers)

Stem Leaf
1 0,5,7
2 3,5
3 2,4,6
4 1,1,3,8
5 2,2,4,5,7
6 0,0,1,2,3,4,4,5,7,7,9
7 0,1,2,4,4,4,5,6,6,7,8,8,9,9
8 0,0,0,1,1,2,2,3,4,4,5,5,8,9
9 0,2,5,8

The stem-and-leaf plot is complete. The last thing to do really is to write a key or legend at the bottom of the table to give the reader a way to read the table.
For instance, you can write 2 | 3 to mean the number 23. The vertical line separates the stem from the leaf.

===========================================================================

Part (b)

As mentioned earlier, we have n = 60 values.

n/2 = 60/2 = 30
The median is the midpoint of the values in slot 30 and slot 31.
Those values are 71 and 72. It might help to break the 6 by 10 block into two 3 by 10 blocks to quickly locate 71 and 72.

Compute the midpoint: (71+72)/2 = 71.5

Median = 71.5

The command named "MEDIAN" in spreadsheet software will quickly determine the median.

Now split the data into two halves
L = lower half, stuff smaller than median
U = upper half, stuff larger than median

L = {
10,15,17,23,25,32,34,36,41,41
43,48,52,52,54,55,57,60,60,61
62,63,64,64,65,67,67,69,70,71
}

U = {
72,74,74,74,75,76,76,77,78,78
79,79,80,80,80,81,81,82,82,83
84,84,85,85,88,89,90,92,95,98
}

Find the median of set L. That value is 54.5. This is the value of Q1 (aka first quartile).

The value of Q3 is 80.5 because it is the median of set U.

Summary:
lower quartile = Q1 = 54.5
Median = 71.5
upper quartile = Q3 = 80.5

===========================================================================

Part (c)

Use the spreadsheet function named COUNTIF to count the number of values smaller than 65.

The command I typed in was: =COUNTIF(A1:J6,"<65") where A1 is the top left corner of the block of data values.
The portion A1:J6 means "look at the rectangle that has A1 in the top left corner and J6 in the bottom right corner".
Don't forget about the equal sign up front.

The result of that command is 24.

Here is a list of those 24 values that are smaller than 65:

10 15 17 23 25 32
34 36 41 41 43 48
52 52 54 55 57 60
60 61 62 63 64 64

This table has 4 rows and 6 columns.

There are 24 values we want out of 60 values total.

24/60 = 0.40 = 40% of the scores are less than 65.

Answer: 40%