Question

The following data represent the lifetimes (in months) of a sample of eleven bulbs:

• A. The sample mean is greater than the sample median
• B. The mean deviation about the point \(15\) equals \(3\)
• C. The range of the data is twice the interquartile range of the data
• D. The sample standard deviation as well as the sample median are scaled by \(2\) if every data point is scaled by \(2\)

Hint:

When every observation is multiplied by a constant \(c\), the mean, median, and standard deviation are also multiplied by \(c\), while variance is multiplied by \(c^2\).

Answer 1

The Correct Option is A, B, D

Step 1: Order the data.
Sorted: $10,10,12,12,13,14,15,15,17,20,22$.

Step 2: Mean versus median (A).
The mean is $\frac{160}{11}\approx14.55$ and the median (6th value) is $14$, so the mean is larger. (A) holds.

Step 3: Mean deviation about $15$ (B).
The absolute gaps from $15$ sum to $0+5+3+5+2+3+1+5+2+7+0=33$, so the mean deviation is $\frac{33}{11}=3$. (B) holds.

Step 4: Range versus IQR (C).
Range $=22-10=12$, while $Q_1=12$, $Q_3=17$ give $IQR=5$, so twice the IQR is $10\ne12$. (C) fails.

Step 5: Scaling (D).
Multiplying all data by $2$ multiplies both the median and the standard deviation by $2$. (D) holds.
\[ \boxed{(A),(B),(D)} \]