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