To determine the missing data value, we start with the given information:
- The mean (μ) of the data set is 120.
- The standard deviation (σ) of the data set is 9.
The data value in question is known to be exactly 3 standard deviations away from the mean. In statistical terms, a value that is \( k \) standard deviations away from the mean can be calculated using the formula:
[tex]\[ \text{value} = \mu \pm k\sigma \][/tex]
Here, \( k \) is given as 3. So, we need to consider both positive and negative deviations:
1. Positive Deviation:
[tex]\[
\text{value}_1 = \mu + 3\sigma
\][/tex]
Substituting the values, we get:
[tex]\[
\text{value}_1 = 120 + 3 \times 9 = 120 + 27 = 147
\][/tex]
2. Negative Deviation:
[tex]\[
\text{value}_2 = \mu - 3\sigma
\][/tex]
Substituting the values, we get:
[tex]\[
\text{value}_2 = 120 - 3 \times 9 = 120 - 27 = 93
\][/tex]
Therefore, the two possible missing data values are 147 and 93. To match the possible values given in the options:
- 129
- 147
- 360
- 369
We see that 147 is the correct missing data value.
So, the correct answer is:
[tex]\[
\boxed{147}
\][/tex]
