If a sample has a mean of 100 and a standard deviation of 6, what is the value in the set that corresponds to a z-score of 2?
The Correct Answer and Explanation is:
To find the value in the set that corresponds to a z-score of 2, we can use the formula for calculating a z-score:
[
z = \frac{(X – \mu)}{\sigma}
]
Where:
- ( z ) is the z-score,
- ( X ) is the value in the set,
- ( \mu ) is the mean of the set, and
- ( \sigma ) is the standard deviation of the set.
Given:
- ( z = 2 )
- ( \mu = 100 )
- ( \sigma = 6 )
We can rearrange the formula to solve for ( X ):
[
X = z \cdot \sigma + \mu
]
Substituting the known values into the equation:
[
X = 2 \cdot 6 + 100
]
Calculating:
[
X = 12 + 100 = 112
]
Thus, the value in the set that corresponds to a z-score of 2 is 112.
Explanation of Z-Scores
A z-score measures how many standard deviations an individual data point (X) is from the mean ((\mu)) of the dataset. A z-score of 0 indicates that the data point is exactly at the mean, while positive z-scores indicate values above the mean, and negative z-scores indicate values below the mean.
In our example, a z-score of 2 means that the value is two standard deviations above the mean. This is significant in statistics because it helps to identify how unusual or extreme a particular value is compared to the rest of the data. In many applications, especially in hypothesis testing and confidence intervals, understanding the z-score is crucial for interpreting data distributions.
The standard deviation provides insight into the variability of the data. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation suggests more spread-out values. By incorporating these concepts, we can understand the context and relevance of the calculated value, which in this case is 112. It signifies that this value is relatively high in the distribution, representing a point where a significant proportion of the data lies below it.