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A scientist calculated the mean and standard deviation of a data set to be [tex]\mu = 120[/tex] and [tex]\sigma = 9[/tex]. She then found that she was missing one data value from the set. She knows that the missing data value was exactly 3 standard deviations away from the mean. What was the missing data value?

A. 129
B. 147
C. 360
D. 369



Answer :

GinnyAnswer
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]

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