Question

1. for the following set of scores: 7 5 6 4 4 3 8 9 4 7 5 5 6 9 4 7 5 10 6 8 5 6 3 4 8 5 a. construct a frequency distribution table. b. sketch a histogram showing the distribution. c. what is the shape of the distribution?

Explanation:

Part a: Construct a Frequency Distribution Table

Step 1: Identify the Scores and Their Range

First, we list out all the scores and find the minimum and maximum values. The scores are: 7, 5, 6, 4, 4, 3, 8, 9, 4, 7, 5, 5, 6, 9, 4, 7, 5, 10, 6, 8, 5, 6, 3, 4, 8, 5. The minimum score is 3 and the maximum score is 10.

Step 2: Count the Frequency of Each Score

We go through each score and count how many times each score appears:

• Score 3: Appears 2 times (from the list: 3, 3)
• Score 4: Appears 5 times (4, 4, 4, 4, 4)
• Score 5: Appears 7 times (5, 5, 5, 5, 5, 5, 5)
• Score 6: Appears 5 times (6, 6, 6, 6, 6)
• Score 7: Appears 3 times (7, 7, 7)
• Score 8: Appears 3 times (8, 8, 8)
• Score 9: Appears 2 times (9, 9)
• Score 10: Appears 1 time (10)

Step 3: Construct the Frequency Distribution Table

We organize the scores (X) and their frequencies (f) into a table:

Score (X)	Frequency (f)
4	5
5	7
6	5
7	3
8	3
9	2
10	1

Part b: Sketch a Histogram

To sketch the histogram:

1. X - axis (Horizontal Axis): Represent the scores (3, 4, 5, 6, 7, 8, 9, 10).
2. Y - axis (Vertical Axis): Represent the frequency.
3. Bars: For each score, draw a bar whose height is equal to the frequency of that score. For example:

• For score 3, the bar height is 2.
• For score 4, the bar height is 5.
• For score 5, the bar height is 7 (the tallest bar).
• For score 6, the bar height is 5.
• For score 7, the bar height is 3.
• For score 8, the bar height is 3.
• For score 9, the bar height is 2.
• For score 10, the bar height is 1 (the shortest bar).

The histogram will have bars centered on each score (since the scores are discrete) with heights corresponding to their frequencies.

Part c: Shape of the Distribution

To determine the shape, we analyze the frequency distribution:

• The score with the highest frequency is 5 (frequency = 7).
• The frequencies decrease as we move away from 5 in both directions (left: 3, 4; right: 6, 7, 8, 9, 10).
• The left - hand side (lower scores: 3, 4) and the right - hand side (higher scores: 6, 7, 8, 9, 10) of the peak (score 5) are roughly symmetric in terms of the pattern of frequency decrease. The frequencies for 3 (2) and 9 (2) are equal, for 4 (5) and 6 (5) are equal, for 7 (3) and 8 (3) are equal, and then 10 has a frequency of 1. So the distribution is approximately symmetric (more specifically, it is a bell - shaped or approximately normal - like distribution, although it is a discrete distribution).

Final Answers

a. The frequency distribution table is as shown above.
b. The histogram is constructed with scores on the x - axis and frequencies on the y - axis, with bar heights equal to the frequencies of each score.
c. The shape of the distribution is approximately symmetric (bell - shaped).

Answer:

Part a: Construct a Frequency Distribution Table

Step 1: Identify the Scores and Their Range

First, we list out all the scores and find the minimum and maximum values. The scores are: 7, 5, 6, 4, 4, 3, 8, 9, 4, 7, 5, 5, 6, 9, 4, 7, 5, 10, 6, 8, 5, 6, 3, 4, 8, 5. The minimum score is 3 and the maximum score is 10.

Step 2: Count the Frequency of Each Score

We go through each score and count how many times each score appears:

• Score 3: Appears 2 times (from the list: 3, 3)
• Score 4: Appears 5 times (4, 4, 4, 4, 4)
• Score 5: Appears 7 times (5, 5, 5, 5, 5, 5, 5)
• Score 6: Appears 5 times (6, 6, 6, 6, 6)
• Score 7: Appears 3 times (7, 7, 7)
• Score 8: Appears 3 times (8, 8, 8)
• Score 9: Appears 2 times (9, 9)
• Score 10: Appears 1 time (10)

Step 3: Construct the Frequency Distribution Table

We organize the scores (X) and their frequencies (f) into a table:

Score (X)	Frequency (f)
4	5
5	7
6	5
7	3
8	3
9	2
10	1

Part b: Sketch a Histogram

To sketch the histogram:

1. X - axis (Horizontal Axis): Represent the scores (3, 4, 5, 6, 7, 8, 9, 10).
2. Y - axis (Vertical Axis): Represent the frequency.
3. Bars: For each score, draw a bar whose height is equal to the frequency of that score. For example:

• For score 3, the bar height is 2.
• For score 4, the bar height is 5.
• For score 5, the bar height is 7 (the tallest bar).
• For score 6, the bar height is 5.
• For score 7, the bar height is 3.
• For score 8, the bar height is 3.
• For score 9, the bar height is 2.
• For score 10, the bar height is 1 (the shortest bar).

The histogram will have bars centered on each score (since the scores are discrete) with heights corresponding to their frequencies.

Part c: Shape of the Distribution

To determine the shape, we analyze the frequency distribution:

• The score with the highest frequency is 5 (frequency = 7).
• The frequencies decrease as we move away from 5 in both directions (left: 3, 4; right: 6, 7, 8, 9, 10).
• The left - hand side (lower scores: 3, 4) and the right - hand side (higher scores: 6, 7, 8, 9, 10) of the peak (score 5) are roughly symmetric in terms of the pattern of frequency decrease. The frequencies for 3 (2) and 9 (2) are equal, for 4 (5) and 6 (5) are equal, for 7 (3) and 8 (3) are equal, and then 10 has a frequency of 1. So the distribution is approximately symmetric (more specifically, it is a bell - shaped or approximately normal - like distribution, although it is a discrete distribution).

Final Answers

a. The frequency distribution table is as shown above.
b. The histogram is constructed with scores on the x - axis and frequencies on the y - axis, with bar heights equal to the frequencies of each score.
c. The shape of the distribution is approximately symmetric (bell - shaped).