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Question #8
Given that the median is 270 and the interquartile range is 20 , which of the following statements are true?
I. Fifty percent of the data are greater than or equal to 270 .
II. Fifty percent of the data are between 260 and 280.
III. Seventy-five percent of the data are less than or equal to 280 .
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I, II, and III

Question \# $8$ $\newline$ Given that the median is $270$ and the interquartile range is $20$ , which of the following statements are true? $\newline$ I. Fifty percent of the data are greater than or equal to $270$ . $\newline$ II. Fifty percent of the data are between $260$ and $280$ . $\newline$ III. Seventy-five percent of the data are less than or equal to $280$ . $\newline$ (A) I only $\newline$ (B) II only $\newline$ (C) III only $\newline$ (D) I and II $\newline$ (E) I, II, and III

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Interpret confidence intervals for population means

Full solution

Q. Question \#

8

\newline

Given that the median is

270

and the interquartile range is

20

, which of the following statements are true?

\newline

I. Fifty percent of the data are greater than or equal to

270

.

\newline

II. Fifty percent of the data are between

260

and

280

.

\newline

III. Seventy-five percent of the data are less than or equal to

280

.

\newline

(A) I only

\newline

(B) II only

\newline

(C) III only

\newline

(D) I and II

\newline

(E) I, II, and III

Understand Median and IQR: Step $1$ : Understand the definition of median and interquartile range. $\newline$ Median is the middle value of the data set, so $50\%$ of the data are greater than or equal to $270$ and $50\%$ are less than or equal to $270$ . Interquartile range (IQR) is the range between the $25$ th percentile ( $Q1$ ) and the $75$ th percentile ( $Q3$ ) of the data, which is $20$ in this case.
Calculate Q $1$ and Q $3$ : Step $2$ : Calculate Q $1$ and Q $3$ using the median and IQR. $\newline$ $Q1 = \text{Median} - (\text{IQR}/2) = 270 - (20/2) = 270 - 10 = 260$ $\newline$ $Q3 = \text{Median} + (\text{IQR}/2) = 270 + (20/2) = 270 + 10 = 280$ $\newline$ This means $50\%$ of the data lies between $260$ and $280$ .
Analyze Statements: Step $3$ : Analyze the statements based on calculations. $\newline$ Statement I: True, as $50\%$ of the data are greater than or equal to $270$ . $\newline$ Statement II: True, as $50\%$ of the data are between $260$ and $280$ . $\newline$ Statement III: True, as $75\%$ of the data are less than or equal to $280$ (since $Q3$ is $280$ ).

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