Multiple choice

Find the value of $p$ if the mean of the following distribution whose mean is $20$. $x$ 15 17 19 20 + $p$ 23 $f$ 2 3 4 5$p$ 6

  1. $p = 0$
  2. $p = 1$
  3. $p = 2$

  4. Data insufficient
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B Correct answer
AI explanation

Using the direct method for the arithmetic mean, the sum of observations is (15 times 2) plus (17 times 3) plus (19 times 4) plus (20 plus p)(5p) plus (23 times 6), which equals 279 plus 100p plus 5p squared. The sum of frequencies is 2 plus 3 plus 4 plus 5p plus 6, which equals 15 plus 5p. Setting the mean to 20 gives (279 plus 100p plus 5p squared) divided by (15 plus 5p) equals 20. Multiplying and simplifying gives 5p squared equals 21p plus 21, so p equals 1.

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