To determine the relationship between the score of 694 and the mean using standard deviations, we need to follow these steps:
1. Identify the Given Values:
- Mean, [tex]\(\mu = 690\)[/tex]
- Standard deviation, [tex]\(\sigma = 14\)[/tex]
- Specific score, [tex]\(X = 694\)[/tex]
2. Calculate the Z-score:
The Z-score formula is given by:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Plugging in the values:
[tex]\[
z_{694} = \frac{694 - 690}{14}
\][/tex]
[tex]\[
\frac{4}{14} = 0.2857142857142857
\][/tex]
3. Interpret the Z-score:
A Z-score of [tex]\(0.2857142857142857\)[/tex] indicates how many standard deviations the specific score (694) is from the mean (690).
4. Determine the Category:
- If the Z-score is within [tex]\(1\)[/tex] standard deviation (i.e., [tex]\(-1 \leq z \leq 1\)[/tex]), the first statement is true.
- If the Z-score is between [tex]\(1\)[/tex] and [tex]\(2\)[/tex] standard deviations (i.e., [tex]\(1 < |z| \leq 2\)[/tex]), the second statement is true.
- If the Z-score is between [tex]\(2\)[/tex] and [tex]\(3\)[/tex] standard deviations (i.e., [tex]\(2 < |z| \leq 3\)[/tex]), the third statement is true.
- If the Z-score is more than [tex]\(3\)[/tex] standard deviations, the fourth statement is true.
Given that [tex]\(z_{694} = 0.2857142857142857\)[/tex], it clearly falls within the range of [tex]\(-1 \leq z \leq 1\)[/tex], indicating that it is within 1 standard deviation of the mean.
Conclusion:
The statement that must be true about [tex]\(z_{694}\)[/tex] is:
- [tex]\(z_{694}\)[/tex] is within 1 standard deviation of the mean.
