Answer :
To address the student's question, we need to use the empirical rule, which applies to bell-shaped (normal) distributions. The empirical rule states:
- Approximately 68% of data falls within one standard deviation of the mean.
- Approximately 95% falls within two standard deviations.
- Approximately 99.7% falls within three standard deviations.
Given that the mean [tex]\mu[/tex] is 35.9 and the standard deviation [tex]\sigma[/tex] is 9.6, let's find the 68% interval:
First, calculate the interval for one standard deviation from the mean:
- Lower bound: [tex]\mu - \sigma = 35.9 - 9.6 = 26.3[/tex]
- Upper bound: [tex]\mu + \sigma = 35.9 + 9.6 = 45.5[/tex]
So, the 68% interval is between 26.3 and 45.5.
Next, for the usual range:
According to the empirical rule (or the typical definition of the usual range), it's often interpreted as the range in which 95% of data falls.
- Lower bound: [tex]\mu - 2\sigma = 35.9 - 2(9.6) = 35.9 - 19.2 = 16.7[/tex]
- Upper bound: [tex]\mu + 2\sigma = 35.9 + 2(9.6) = 35.9 + 19.2 = 55.1[/tex]
Therefore, the usual range is between 16.7 and 55.1.
In conclusion:
- 68% interval: From 26.3 to 45.5
- Usual range (95% of data): From 16.7 to 55.1