assume that a set of test score is normally distrubuted with a meanof 100 and a standard deviation of 20 find the quantities using the 68-95-99.7 rule to get:
the percentage of scores less than 100:
B) relative frequency of scores less than 120 c) percentage ofless than 140, D)percentage of less than 80, relative frequency of score less than 60,F) pertenage of scores greater than 120?
A. In normal distribution, mean = median
Z = (score - mean)/SD
B. Z = +1, p = mean + 1/2 (68)
Use similar process for the remainder of the questions.
To solve this problem, we will use the 68-95-99.7 rule, also known as the empirical rule, which tells us the percentage of data points that fall within a certain number of standard deviations from the mean in a normal distribution.
Before we can use the 68-95-99.7 rule, we need to convert the scores to z-scores. The z-score is a measure of how many standard deviations a data point is away from the mean.
The formula to calculate the z-score is: z = (x - mean) / standard deviation
Let's calculate the z-scores for each of the given values:
A) Percentage of scores less than 100:
To find the percentage of scores less than 100, we need to calculate the z-score for 100. Since the mean is 100 and the standard deviation is 20, the z-score is:
z = (100 - 100) / 20
z = 0
Since the z-score is 0, this means that the score of 100 is exactly at the mean. According to the empirical rule, 50% of the distribution falls below the mean. Therefore, the percentage of scores less than 100 is 50%.
B) Relative frequency of scores less than 120:
To find the relative frequency of scores less than 120, we need to calculate the z-score for 120:
z = (120 - 100) / 20
z = 1
According to the empirical rule, approximately 68% of the distribution falls within one standard deviation of the mean, and since the z-score is 1, which is within one standard deviation, the relative frequency of scores less than 120 is also approximately 68%.
C) Percentage of scores less than 140:
To find the percentage of scores less than 140, we need to calculate the z-score for 140:
z = (140 - 100) / 20
z = 2
According to the empirical rule, approximately 95% of the distribution falls within two standard deviations of the mean, and since the z-score is 2, which is within two standard deviations, the percentage of scores less than 140 is approximately 95%.
D) Percentage of scores less than 80:
To find the percentage of scores less than 80, we need to calculate the z-score for 80:
z = (80 - 100) / 20
z = -1
According to the empirical rule, approximately 68% of the distribution falls within one standard deviation of the mean, and since the z-score is -1, which is within one standard deviation, the percentage of scores less than 80 is also approximately 68%.
E) Relative frequency of scores less than 60:
To find the relative frequency of scores less than 60, we need to calculate the z-score for 60:
z = (60 - 100) / 20
z = -2
According to the empirical rule, approximately 95% of the distribution falls within two standard deviations of the mean, and since the z-score is -2, which is within two standard deviations, the relative frequency of scores less than 60 is approximately 95%.
F) Percentage of scores greater than 120:
To find the percentage of scores greater than 120, we need to subtract the percentage of scores less than 120 from 100%:
Percentage of scores greater than 120 = 100% - 68%
Percentage of scores greater than 120 = 32%
Therefore, the percentage of scores greater than 120 is 32%.
I hope this helps clarify how to use the 68-95-99.7 rule to find the given percentages and relative frequencies!