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Select the outlier in the data set. $\newline$ $20, 30, 34, 61, 62, 90, 98, 566$ $\newline$ If the outlier were removed from the data set, would the mean increase or decrease? $\newline$ Choices: $\newline$ (A)increase $\newline$ (B)decrease

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Identify an outlier and describe the effect of removing it

Full solution

Q. Select the outlier in the data set.

\newline

20, 30, 34, 61, 62, 90, 98, 566

\newline

If the outlier were removed from the data set, would the mean increase or decrease?

\newline

Choices:

\newline

(A)increase

\newline

(B)decrease

Arrange and Calculate Mean: Arrange the data set in ascending order and calculate the mean. $\newline$ The data set in ascending order: $20, 30, 34, 61, 62, 90, 98, 566$ . $\newline$ Mean $= (20 + 30 + 34 + 61 + 62 + 90 + 98 + 566) / 8 = 961 / 8 = 120.125$
Calculate IQR and Identify Outliers: Calculate the interquartile range (IQR) to identify outliers. $\newline$ First, find the median (Q $2$ ), which is the average of the $4$ th and $5$ th values: $(61 + 62) / 2 = 61.5$ $\newline$ Next, find Q $1$ , the median of the first half of the data: $(30 + 34) / 2 = 32$ $\newline$ Then, find Q $3$ , the median of the second half of the data: $(90 + 98) / 2 = 94$ $\newline$ IQR = Q $3$ - Q $1$ = $94$ - $32$ = $62$
Determine Outlier Threshold: Determine the outlier threshold. $\newline$ Lower bound = $Q1 - 1.5 \times IQR = 32 - 1.5 \times 62 = 32 - 93 = -61$ (since there can't be a negative value in this context, we'll consider the lower bound as the smallest value in the data set) $\newline$ Upper bound = $Q3 + 1.5 \times IQR = 94 + 1.5 \times 62 = 94 + 93 = 187$ $\newline$ Any value outside of the range $[-61, 187]$ is considered an outlier.
Identify Outlier: Identify the outlier. $\newline$ The value $566$ is outside the range $[-61, 187]$ , so it is the outlier.
Determine Mean Change: Determine if the mean would increase or decrease if the outlier were removed. $\newline$ Removing the outlier $566$ would decrease the sum of the data set without changing the number of values significantly. Therefore, the mean would decrease.

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