Sketch graphs of the distributions of the scores in Jake's class and in Elena's class.
Elena, see solution.
Practice makes perfect
We are told that both Jake and Elena scored 87 on the same standardized test. Also, we are told that they are in different study groups. We are asked to determine if either of them scored in the top 10 % of their group. Let's start with Jake.
Jake
Let's make a graph of the distribution of Jake's group's test scores. In his group, the mean was 80 points and the standard deviation was 6. First, let's find the values that are one, two, and three standard deviations away from the mean. For convenience, the mean will be represented by the letter m and the standard deviation will be represented by s.
m-3s
m-2s
m-s
m
m+s
m+2s
m+3s
Substitute
80-3( 6)
80-2( 6)
80- 6
80
80+ 6
80+2( 6)
80+3( 6)
Simplify
62
68
74
80
86
92
98
Now, let's draw vertical lines with the calculated values.
Now, we are ready to sketch the normal curve. Let's draw a bell-shaped curve with its highest point at the mean, 80.
The normal curve is divided into sections of standard deviation widths. Let's label the percentages of each section.
Jake scored 87 on the test. Therefore, the scores lower than Jake's are the ones with 86 points or lower. Let's highlight them on the graph.
Let's add up all the highlighted percentages.
0.15 %+2.35 %+13.5 %+34 %+34 %=84 %
We found that 84 % percent of the people in Jake's group scored lower than him. Therefore, Jake's score is in the top 100 %-84 %=16 % of his group. This means that Jake's score is not in the top 10 % of his class.
Elena
In Elena's class, the mean of the scores is 76 and the standard deviation is 4. Let's make a graph of the distribution of the scores.
Just like before, let's label the percentages of each section.
Elena scored 87 points on the test. Thus, the scores higher than her score are the ones with at least 88 points. Let's highlight them.
Only 0.15 % of the scores are higher than Elena's score! This means that Elena's score must be in the top 10 % of her class.
Conclusion
We found that only Elena's score is in the top 10 % of her class. Jake's score is in the top 16 % of his class.
