Multiple choice

Let $x_{1},\ x_{2},....x_{n}$ be $n$ observation. Let $w_{1}=l_{x_{1}}+k$ for $i=1.2,..n$, where $l$ and $k$ are constants. If the mean of $x_{1}s$ is $48$ and their standard deviation is $12$, the mean of $w_{1}s$ is $55$ and standard deviation of $w_{1}s$ is $15$. The value of $l$ and $k$ should be

  1. $l=1.25,\ k=-5$
  2. $l=-1.25,\ k=5$
  3. $l=2.5,\ k=-5$
  4. $l=2.5,\ k=5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For w = lx + k, the standard deviation becomes |l| times the original standard deviation. Thus |l| = 15/12 = 1.25, and using the means gives 55 = 1.25 × 48 + k, so k = -5. The positive value l = 1.25 gives option A.

AI explanation

The new standard deviation equals the original standard deviation multiplied by the scale factor l, so 15 equals 12 times l, which gives l = 1.25. The new mean equals the original mean multiplied by l plus the constant k, so 55 equals 48 times 1.25 plus k, yielding k = -5. Therefore, l is 1.25 and k is -5.

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