To determine the value of \(k - 2p\), it is necessary to first ascertain the values of \(k\) and \(p\).
Step 1: Determine \(k\)
The provided set of numbers is 3, 4, 9, 2k, 10, 8, 6, and \(k + 6\). Their mean is 8. The calculation of the mean is defined as:
\(\text{Mean} = \frac{\text{Sum of all elements}}{\text{Number of elements}}\)
Applying this formula to the given numbers yields:
\(\frac{3+4+9+2k+10+8+6+(k+6)}{8} = 8\)
Simplifying this equation to solve for \(k\):
\(46 + 3k = 64\)
\(3k = 64 - 46\)
\(3k = 18\)
\(k = \frac{18}{3}\)
\(k = 6\)
Step 2: Determine \(p\)
The set of numbers is 2, 2, 3, \(2p\), \(2p + 1\), 4, 4, 5, and 6. The mode is stated to be 4. The mode represents the value that occurs with the highest frequency.
For 4 to be the mode, its frequency must exceed that of any other number. Analyzing the possible values of \(2p\):
| If \(2p = 4\) | This implies \(p = 2\), and the sequence becomes 2, 2, 3, 4, 5, 4, 4, 5, 6. In this case, 4 appears 3 times, which is the highest frequency. |
| If \(2p eq 4\) | For instance, if \(2p = 3\), the sequence would be 2, 2, 3, 3, 4, 4, 5, 6. Here, 2 and 3 would both have a frequency of 2, and 4 would also have a frequency of 2, leading to multiple modes or a different mode if \(2p\) made another number more frequent. Therefore, the mode would not be uniquely 4. |
Given that 4 is the mode, it must be that \(2p = 4\), which leads to \(p = 2\).
Step 3: Compute \(k - 2p\)
Using the determined values of \(k = 6\) and \(p = 2\), the calculation is as follows:
\(k - 2p = 6 - 2 \times 2\)
\(= 6 - 4\)
\(= 2\)
Consequently, the value of \(k - 2p\) is 2.