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Select the outlier in the data set. $\newline$ $4, 55, 57, 58, 61, 63, 68, 72, 82$ $\newline$ 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

4, 55, 57, 58, 61, 63, 68, 72, 82

\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 Data Set: Arrange the data set in ascending order and identify any value that is significantly different from the others. $\newline$ The data set in ascending order is: $4, 55, 57, 58, 61, 63, 68, 72, 82$ . $\newline$ The value $4$ stands out as being significantly lower than the rest of the values.
Calculate IQR: Calculate the interquartile range (IQR) to determine the outlier threshold. $\newline$ First, find the first quartile ( $Q_1$ ) and the third quartile ( $Q_3$ ) of the data set. $\newline$ $Q_1$ is the median of the first half: $(55 + 57) / 2 = 56$ . $\newline$ $Q_3$ is the median of the second half: $(68 + 72) / 2 = 70$ . $\newline$ $IQR = Q_3 - Q_1 = 70 - 56 = 14$ .
Calculate Outlier Boundaries: Calculate the outlier boundaries using the IQR. $\newline$ Lower boundary = $Q1 - 1.5 \times IQR = 56 - 1.5 \times 14 = 56 - 21 = 35$ . $\newline$ Upper boundary = $Q3 + 1.5 \times IQR = 70 + 1.5 \times 14 = 70 + 21 = 91$ . $\newline$ Any value below $35$ or above $91$ is considered an outlier.
Identify Outlier: Identify the outlier based on the calculated boundaries. $\newline$ The value $4$ is below the lower boundary of $35$ , so it is an outlier.
Effect on Mean: Determine the effect on the mean if the outlier is removed. $\newline$ Removing a value that is lower than the mean will result in an increase in the mean. $\newline$ Since $4$ is lower than the rest of the values, removing it will increase the mean.

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