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Answer :
When dealing with a normally distributed random variable [tex]\( X \)[/tex], we can use the empirical rule, which is a statistical rule describing the distribution of data.
Here's a step-by-step breakdown:
1. Normal Distribution Basics:
- A normal distribution is symmetric, with most of the data clustering around a central point — the mean.
- The spread or dispersion of the distribution is measured by the standard deviation.
2. Empirical Rule (68-95-99.7 Rule):
- This rule helps us understand how data is spread in a normal distribution.
- 68% of the data falls within 1 standard deviation ([tex]\( \sigma \)[/tex]) of the mean ([tex]\( \mu \)[/tex]).
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of the data falls within 3 standard deviations of the mean.
3. Within 1 Standard Deviation:
- For this specific question, we are interested in the percentage of data within 1 standard deviation of the mean.
- According to the empirical rule, approximately 68% of the data values lie within 1 standard deviation above and below the mean.
Thus, the percentage of observed values of [tex]\( x \)[/tex] that fall within 1 standard deviation of the mean in a normal distribution is 68%.
Here's a step-by-step breakdown:
1. Normal Distribution Basics:
- A normal distribution is symmetric, with most of the data clustering around a central point — the mean.
- The spread or dispersion of the distribution is measured by the standard deviation.
2. Empirical Rule (68-95-99.7 Rule):
- This rule helps us understand how data is spread in a normal distribution.
- 68% of the data falls within 1 standard deviation ([tex]\( \sigma \)[/tex]) of the mean ([tex]\( \mu \)[/tex]).
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of the data falls within 3 standard deviations of the mean.
3. Within 1 Standard Deviation:
- For this specific question, we are interested in the percentage of data within 1 standard deviation of the mean.
- According to the empirical rule, approximately 68% of the data values lie within 1 standard deviation above and below the mean.
Thus, the percentage of observed values of [tex]\( x \)[/tex] that fall within 1 standard deviation of the mean in a normal distribution is 68%.
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