Question

If the mean of 1,3,4,5,7,4 is \(m\) and mean of 3,2,2,4,3,3,\(p\) is \((m-1)\) and median is \(q\) then \( p+q = \)

• A. 4
• B. 5
• C. 6
• D. 7 , leads to 7

Hint:

1. Calculate mean \(m\) of first set: \(m = (1+3+4+5+7+4)/6 = 24/6 = 4\). 2. Mean of second set is \(m-1 = 4-1 = 3\). 3. Second set: 3,2,2,4,3,3,\(p\). Sum = \(17+p\). Count = 7. So, \((17+p)/7 = 3 \implies 17+p = 21 \implies p=4\). 4. Second set becomes: 3,2,2,4,3,3,4. Ordered: 2,2,3,3,3,4,4. Median \(q\) (middle value of 7 terms) is the 4th term, which is 3. 5. \(p+q = 4+3 = 7\).

Answer 1

The Correct Option is D

Concept:
Mean = (Sum of observations) / (Number of observations).
Median = Middle value of a dataset when arranged in order. If there are an even number of observations, the median is the average of the two middle values. If odd, it's the single middle value. The term "median is \(q\)" is assumed to refer to the median of the second dataset. Step 1: Calculate \(m\) (mean of the first dataset) First dataset: 1, 3, 4, 5, 7, 4. Number of observations (\(N_1\)) = 6. Sum of observations (\(S_1\)) = \(1+3+4+5+7+4 = 24\). Mean \(m = \frac{S_1}{N_1} = \frac{24}{6} = 4\). Step 2: Determine the mean of the second dataset The mean of the second dataset is given as \((m-1)\). Since \(m=4\), the mean of the second dataset is \(4-1 = 3\). Step 3: Calculate \(p\) using the mean of the second dataset Second dataset: 3, 2, 2, 4, 3, 3, \(p\). Number of observations (\(N_2\)) = 7. Sum of known observations in the second dataset = \(3+2+2+4+3+3 = 17\). Sum of all observations in the second dataset (\(S_2\)) = \(17+p\). Mean of second dataset = \(\frac{S_2}{N_2} = \frac{17+p}{7}\). We know this mean is 3: \[ \frac{17+p}{7} = 3 \] Multiply by 7: \(17+p = 3 \times 7 = 21\). \[ p = 21 - 17 = 4 \] So, \(p=4\). Step 4: Find \(q\) (median of the second dataset) The second dataset, with \(p=4\), is: 3, 2, 2, 4, 3, 3, 4. To find the median, first arrange the data in ascending order: 2, 2, 3, 3, 3, 4, 4. There are \(N_2 = 7\) observations (an odd number). The median is the middle value, which is the \(\left(\frac{N_2+1}{2}\right)^{th}\) term. Median position = \(\left(\frac{7+1}{2}\right)^{th} = \left(\frac{8}{2}\right)^{th} = 4^{th}\) term. The 4th term in the ordered dataset (2, 2, 3, 3, 3, 4, 4) is 3. So, the median \(q = 3\). Step 5: Calculate \(p+q\) We found \(p=4\) and \(q=3\). \[ p+q = 4+3 = 7 \]