Answer :
To determine whether the statement "A z-score of 1.0 corresponds to a value one standard deviation from the mean" is true or false, let's break down the concept of a z-score.
1. Understanding the z-score:
- The z-score is a statistical measure that describes a value's relationship to the mean of a group of values.
- It is expressed in terms of standard deviations from the mean.
- The formula to calculate a z-score for a given value [tex]\( x \)[/tex] is:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
where:
- [tex]\( x \)[/tex] is the value in question,
- [tex]\( \mu \)[/tex] is the mean of the data set,
- [tex]\( \sigma \)[/tex] is the standard deviation of the data set.
2. Interpreting a z-score of 1.0:
- When [tex]\( z = 1.0 \)[/tex], we can rearrange the z-score formula to find [tex]\( x \)[/tex]:
[tex]\[ 1.0 = \frac{x - \mu}{\sigma} \][/tex]
Simplifying this, we get:
[tex]\[ x - \mu = 1.0 \cdot \sigma \][/tex]
Therefore:
[tex]\[ x = \mu + \sigma \][/tex]
3. Conclusion:
- From the above equation, [tex]\( x \)[/tex] is one standard deviation ([tex]\( \sigma \)[/tex]) away from the mean ([tex]\( \mu \)[/tex]).
- This confirms that a z-score of 1.0 indeed corresponds to a value that is one standard deviation away from the mean.
Thus, the statement "A z-score of 1.0 corresponds to a value one standard deviation from the mean" is True.
1. Understanding the z-score:
- The z-score is a statistical measure that describes a value's relationship to the mean of a group of values.
- It is expressed in terms of standard deviations from the mean.
- The formula to calculate a z-score for a given value [tex]\( x \)[/tex] is:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
where:
- [tex]\( x \)[/tex] is the value in question,
- [tex]\( \mu \)[/tex] is the mean of the data set,
- [tex]\( \sigma \)[/tex] is the standard deviation of the data set.
2. Interpreting a z-score of 1.0:
- When [tex]\( z = 1.0 \)[/tex], we can rearrange the z-score formula to find [tex]\( x \)[/tex]:
[tex]\[ 1.0 = \frac{x - \mu}{\sigma} \][/tex]
Simplifying this, we get:
[tex]\[ x - \mu = 1.0 \cdot \sigma \][/tex]
Therefore:
[tex]\[ x = \mu + \sigma \][/tex]
3. Conclusion:
- From the above equation, [tex]\( x \)[/tex] is one standard deviation ([tex]\( \sigma \)[/tex]) away from the mean ([tex]\( \mu \)[/tex]).
- This confirms that a z-score of 1.0 indeed corresponds to a value that is one standard deviation away from the mean.
Thus, the statement "A z-score of 1.0 corresponds to a value one standard deviation from the mean" is True.
