Answer :
To solve this problem, we need to determine the percentage of values that are greater than 310 in a normal distribution with a mean (μ) of 67 and a standard deviation (σ) of 81. We'll use the 68-95-99.7 rule, also known as the empirical rule, which approximates the percentages of values that lie within certain numbers of standard deviations from the mean in a normal distribution.
Here’s the step-by-step solution:
1. Calculate the z-score:
The z-score tells us how many standard deviations a particular value (in this case, 310) is from the mean. The formula for the z-score is:
[tex]\[
z = \frac{(X - \mu)}{\sigma}
\][/tex]
Where:
- [tex]\( X \)[/tex] is the value (310 in this case),
- [tex]\( \mu \)[/tex] is the mean (67), and
- [tex]\( \sigma \)[/tex] is the standard deviation (81).
Substituting the values into the formula, we get:
[tex]\[
z = \frac{(310 - 67)}{81} = \frac{243}{81} = 3.0
\][/tex]
2. Interpret the z-score using the empirical rule:
According to the empirical rule:
- About 68% of the data falls within 1 standard deviation of the mean.
- About 95% of the data falls within 2 standard deviations of the mean.
- About 99.7% of the data falls within 3 standard deviations of the mean.
Since our z-score is 3.0, we are looking at values beyond 3 standard deviations from the mean. From the empirical rule:
- Roughly 99.7% of the values lie within 3 standard deviations of the mean.
- Therefore, about 0.3% of the values lie beyond 3 standard deviations (both tails of the distribution).
- Since we are interested in the values greater than 310, we focus on the upper tail.
3. Calculate the percentage:
- The upper tail beyond 3 standard deviations represents half of the 0.3% (since the normal distribution is symmetrical), which is:
[tex]\[
\frac{0.3\%}{2} = 0.15\%
\][/tex]
4. Conclusion:
Therefore, approximately 0.15% of the values are greater than 310.
So, the correct answer is:
[tex]\[ \text{d) } 0.15\% \][/tex]
Here’s the step-by-step solution:
1. Calculate the z-score:
The z-score tells us how many standard deviations a particular value (in this case, 310) is from the mean. The formula for the z-score is:
[tex]\[
z = \frac{(X - \mu)}{\sigma}
\][/tex]
Where:
- [tex]\( X \)[/tex] is the value (310 in this case),
- [tex]\( \mu \)[/tex] is the mean (67), and
- [tex]\( \sigma \)[/tex] is the standard deviation (81).
Substituting the values into the formula, we get:
[tex]\[
z = \frac{(310 - 67)}{81} = \frac{243}{81} = 3.0
\][/tex]
2. Interpret the z-score using the empirical rule:
According to the empirical rule:
- About 68% of the data falls within 1 standard deviation of the mean.
- About 95% of the data falls within 2 standard deviations of the mean.
- About 99.7% of the data falls within 3 standard deviations of the mean.
Since our z-score is 3.0, we are looking at values beyond 3 standard deviations from the mean. From the empirical rule:
- Roughly 99.7% of the values lie within 3 standard deviations of the mean.
- Therefore, about 0.3% of the values lie beyond 3 standard deviations (both tails of the distribution).
- Since we are interested in the values greater than 310, we focus on the upper tail.
3. Calculate the percentage:
- The upper tail beyond 3 standard deviations represents half of the 0.3% (since the normal distribution is symmetrical), which is:
[tex]\[
\frac{0.3\%}{2} = 0.15\%
\][/tex]
4. Conclusion:
Therefore, approximately 0.15% of the values are greater than 310.
So, the correct answer is:
[tex]\[ \text{d) } 0.15\% \][/tex]
