High School

A set of test scores is normally distributed with a mean of 110 and a standard deviation of 15. Use the 68-95-99.7 rule to find the percentage of scores in each of the following categories:

a. Greater than 110
b. Greater than 125
c. Less than 95
d. Less than 140
e. Less than 80
f. Less than 125
g. Greater than 95
h. Between 95 and 125

The percentage of scores greater than 110 is:

Answer :

The question is about understanding the distribution of test scores in a normal distribution, using the 68-95-99.7 rule, also known as the empirical rule. This rule helps us to understand how the data falls around the mean in a normal distribution.

Given:

  • Mean ([tex]\mu[/tex]) = 110
  • Standard Deviation ([tex]\sigma[/tex]) = 15

The 68-95-99.7 Rule:

  1. Approximately 68% of the data falls within 1 standard deviation ([tex]\sigma[/tex]) from the mean.
  2. Approximately 95% of the data falls within 2 standard deviations.
  3. Approximately 99.7% of the data falls within 3 standard deviations.

Now, let’s address each category:

a. Greater than 110:

The mean is 110, so 50% of the scores are above the mean.

b. Greater than 125:

125 is 1 standard deviation ($110 + 15 = 125$) above the mean. About 68% of scores are between 95 and 125. Therefore, 32% are outside this range. Since the distribution is symmetric, 16% are greater than 125.

c. Less than 95:

Similarly, 95 is 1 standard deviation below the mean. As 32% of the scores are outside the 68% range, 16% of scores are less than 95.

d. Less than 140:

140 is 2 standard deviations above the mean ($110 + 2\times15 = 140$). Thus, about 95% of scores are within this range. So, approximately 95% are less than 140.

e. Less than 80:

80 is 2 standard deviations below the mean ($110 - 15\times2 = 80$). About 95% of scores fall within 2 standard deviations of the mean. Hence, the percentage of scores less than 80 is 2.5% because the remaining 5% is symmetric, split equally in both tails.

f. Less than 125:

Since 125 is 1 standard deviation above the mean, 50% (less than the mean) + 34% (from mean to 125) = 84%.

g. Greater than 95:

95 is 1 standard deviation below the mean. Therefore, 84% of scores are greater than 95, calculated as 50% (above mean) + 34% (from 95 to mean).

h. Between 95 and 125:

This range covers scores within 1 standard deviation from the mean in both directions (lower: mean - [tex]\sigma[/tex]; upper: mean + [tex]\sigma[/tex]). As per the empirical rule, about 68% of scores fall between these two values.

In conclusion, understanding the normal distribution and utilizing the empirical rule allows us to estimate the probability of where scores lie within a given data set.