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?
Answer (2)
The problem states that the missing value is 3 standard deviations away from the mean.
Calculate the value 3 standard deviations above the mean: 120 + 3 ( 9 ) = 147 .
Calculate the value 3 standard deviations below the mean: 120 − 3 ( 9 ) = 93 .
The missing value is either 147 or 93, and since 147 is among the options, the answer is 147 .
Explanation
Understanding the Problem We are given that the mean of a data set is μ = 120 and the standard deviation is σ = 9 . We also know that a missing data value is exactly 3 standard deviations away from the mean. Our goal is to find this missing value.
Finding Possible Values The missing value can be either 3 standard deviations above the mean or 3 standard deviations below the mean. Let's calculate both possibilities.
Calculating Value Above the Mean First, we calculate the value 3 standard deviations above the mean: μ + 3 σ = 120 + 3 ( 9 ) = 120 + 27 = 147
Calculating Value Below the Mean Next, we calculate the value 3 standard deviations below the mean: μ − 3 σ = 120 − 3 ( 9 ) = 120 − 27 = 93
Determining the Missing Value Therefore, the missing data value is either 147 or 93. Looking at the multiple-choice options, we see that 147 is one of the options.
Final Answer Thus, the missing data value is 147.
Examples
Understanding standard deviations is crucial in many real-world scenarios. For example, in quality control, manufacturers use standard deviations to ensure that their products meet certain specifications. If a product's measurements deviate too much from the mean (e.g., more than 3 standard deviations), it might indicate a problem in the manufacturing process. Similarly, in finance, standard deviation is used to measure the volatility of investments. A higher standard deviation indicates a riskier investment, as the returns are more likely to deviate significantly from the average return.
The missing data value is either 147 or 93, as it is exactly 3 standard deviations away from the mean of 120. Since 147 is greater than the mean and retains a distance of 3 standard deviations, it stands as a potential answer. Thus, the missing data value is 147.
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