Find the value of p, if the mean of the following distribution is 7.5. x 3 5 7 9 11 13 f 6 8 15 p 8 4
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Find the value of p, if the mean of the following distribution is 7.5. x 3 5 7 9 11 13 f 6 8 15 p 8 4
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Mean = sum(f*x) / sum(f). 7.5 = (18 + 40 + 105 + 9p + 88 + 52) / (6 + 8 + 15 + p + 8 + 4). 7.5 = (303 + 9p) / (41 + p). 307.5 + 7.5p = 303 + 9p. 4.5 = 1.5p, so p = 3.
Using the direct method for the arithmetic mean, we calculate the sum of products of observations and frequencies as (3 times 6) plus (5 times 8) plus (7 times 15) plus (9p) plus (11 times 8) plus (13 times 4), which equals 297 plus 9p. The total frequency is 6 plus 8 plus 15 plus p plus 8 plus 4, which equals 41 plus p. Setting the mean equal to 7.5 gives the equation (297 plus 9p) divided by (41 plus p) equals 7.5. Solving this yields 297 plus 9p equals 307.5 plus 7.5p, so 1.5p equals 10.5, and p equals 3.