Answer :
To solve the problem of finding the percentage of scores less than 48 on a psychology exam that follows a normal distribution, we use the 68-95-99.7 rule, also known as the empirical rule.
Understanding the Problem
- The distribution is normal, with a mean ([tex]\mu[/tex]) of 54 and a standard deviation ([tex]\sigma[/tex]) of 6.
- We want to find the percentage of scores less than 48.
Applying the 68-95-99.7 Rule
The 68-95-99.7 rule helps us understand the distribution of data:
- 68% of the data falls within one standard deviation of the mean ([tex]\mu \pm \sigma[/tex]).
- 95% falls within two standard deviations ([tex]\mu \pm 2\sigma[/tex]).
- 99.7% falls within three standard deviations ([tex]\mu \pm 3\sigma[/tex]).
Step-by-Step Calculation
Calculate the Z-score for 48:
The Z-score measures how many standard deviations a value is from the mean:
[tex]Z = \frac{X - \mu}{\sigma} = \frac{48 - 54}{6} = \frac{-6}{6} = -1[/tex]
Use the Empirical Rule for Z = -1:
Since a Z-score of -1 falls one standard deviation below the mean, the percentage of
values below this Z-score can be derived from the empirical rule.About 16% of scores are below one standard deviation below the mean, because
50% of scores are below the mean and the 68% around the mean is symmetric,
meaning 34% are in the first standard deviation above the mean, leaving 16%
below the first standard deviation beneath the mean.
Conclusion
Therefore, approximately 16% of the scores on the psychology exam were less than 48.
