The mean of 1,7,5,3,4 and 4 is m. The numbers 3,2,4,2,3,3 and p have mean m-1 and median q. Find p and q

Answer

To solve the problem step by step, we will first calculate the mean \( m \) of the given numbers, then use that to find \( p \) and \( q \).

### Step 1: Calculate the mean \( m \) of the numbers 1, 7, 5, 3, 4, and 4.

The formula for mean is:
\[ m = \frac{\text{Sum of observations}}{\text{Total number of observations}} \]

Calculating the sum:
\[ 1 + 7 + 5 + 3 + 4 + 4 = 24 \]

Total number of observations = 6.

Now, substituting into the mean formula:
\[ m = \frac{24}{6} = 4 \]

### Step 2: Use the mean \( m \) to find \( p \).

We know that the mean of the numbers 3, 2, 4, 2, 3, 3, and \( p \) is \( m - 1 \).

Calculating \( m - 1 \):
\[ m - 1 = 4 - 1 = 3 \]

Now, we set up the equation for the mean of the second set of numbers:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} = 3 \]

Calculating the sum of the known numbers:
\[ 3 + 2 + 4 + 2 + 3 + 3 + p = 17 + p \]

Total number of observations = 7.

Setting up the equation:
\[ 3 = \frac{17 + p}{7} \]

### Step 3: Solve for \( p \).

Multiplying both sides by 7:
\[ 21 = 17 + p \]

Now, solving for \( p \):
\[ p = 21 - 17 = 4 \]

### Step 4: Find the median \( q \) of the numbers 3, 2, 4, 2, 3, 3, and \( p \).

First, we need to arrange the numbers in ascending order. The numbers are:
\[ 2, 2, 3, 3, 3, 4, 4 \]

### Step 5: Calculate the median \( q \).

Since there are 7 numbers (an odd number), the median is the middle number, which is given by the formula:
\[ \text{Median} = \text{(n + 1) / 2 th term} \]

Where \( n \) is the number of observations. Here, \( n = 7 \):
\[ \text{Median} = \frac{7 + 1}{2} = \frac{8}{2} = 4 \]

The 4th term in the ordered list is:
\[ 2, 2, 3, 3, 3, 4, 4 \]
The 4th term is 3.

Thus, \( q = 3 \).

### Final Answers:
- \( p = 4 \)
- \( q = 3 \)