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18. From the given bi-variate frequency distribution, determine if there exists any relationship between the advertisement expenditure (in 000 Rs) and sales revenue (in 000 Rs). Also, interpret the result and test the significance of the result. Estimate the sales revenue when the advertisement expenditure is Rs. 4,80,000.

\begin{tabular}{|l|c|c|c|c|c|}
\hline
\begin{tabular}{l}
Sales revenue \\
(in 000 Rs)
\end{tabular} & \multicolumn{4}{|c|}{Advertisement Expenditure (Rs. 000)} \\
\hline
[tex]$0-40$[/tex] & [tex]$0-40$[/tex] & [tex]$40-80$[/tex] & [tex]$80-120$[/tex] & [tex]$120-160$[/tex] & [tex]$160-200$[/tex] \\
\hline
[tex]$0-50$[/tex] & 12 & -2 & - & - & \\
\hline
[tex]$50-100$[/tex] & 6 & 18 & 8 & 1 & \\
\hline
[tex]$100-150$[/tex] & 8 & 4 & 10 & 10 & 1 \\
\hline
[tex]$150-200$[/tex] & - & 5 & 2 & 2 & 2 \\
\hline
[tex]$200-250$[/tex] & & 1 & 4 & 1 & 3 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Sure, let's go through the problem step by step to find the relationship between advertisement expenditure and sales revenue, and then estimate the sales revenue for an advertisement expenditure of Rs. 4,80,000.

### Step 1: Organize the Data
We are given a bi-variate frequency distribution table with sales revenue and advertisement expenditures broken down into intervals. Let's organize this:

#### Sales Revenue (in 000 Rs):
- 0-50
- 50-100
- 100-150
- 150-200
- 200-250

#### Advertisement Expenditure (in 000 Rs):
- 0-40
- 40-80
- 80-120
- 120-160
- 160-200

#### Frequency Table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & 0-40 & 40-80 & 80-120 & 120-160 & 160-200 \\ \hline 0-50 & 12 & -2 & 0 & 0 & 0 \\ \hline 50-100 & 6 & 18 & 8 & 1 & 0 \\ \hline 100-150 & 8 & 4 & 10 & 10 & 1 \\ \hline 150-200 & 0 & 5 & 2 & 2 & 2 \\ \hline 200-250 & 0 & 1 & 4 & 1 & 3 \\ \hline \end{array} \][/tex]

### Step 2: Calculate the Midpoints
For each interval, we will calculate the midpoint, which will represent the average value for that interval.

#### Midpoints for Sales Revenue:
- 0-50: [tex]\( \frac{0+50}{2} = 25 \)[/tex]
- 50-100: [tex]\( \frac{50+100}{2} = 75 \)[/tex]
- 100-150: [tex]\( \frac{100+150}{2} = 125 \)[/tex]
- 150-200: [tex]\( \frac{150+200}{2} = 175 \)[/tex]
- 200-250: [tex]\( \frac{200+250}{2} = 225 \)[/tex]

#### Midpoints for Advertisement Expenditure:
- 0-40: [tex]\( \frac{0+40}{2} = 20 \)[/tex]
- 40-80: [tex]\( \frac{40+80}{2} = 60 \)[/tex]
- 80-120: [tex]\( \frac{80+120}{2} = 100 \)[/tex]
- 120-160: [tex]\( \frac{120+160}{2} = 140 \)[/tex]
- 160-200: [tex]\( \frac{160+200}{2} = 180 \)[/tex]

### Step 3: Create Frequency Distribution
We will create lists of midpoints for sales revenue and advertisement expenditure according to their frequencies.

#### Sales Revenue (SR):
[tex]\[ \text{Sales Revenue Array} = [25, 75, 125, 175, 225] \][/tex]

#### Advertisement Expenditure (AE):
[tex]\[ \text{Advertisement Expenditure Array} = [20, 60, 100, 140, 180] \][/tex]

#### Frequency Distribution:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline & 20 & 60 & 100 & 140 & 180 \\ \hline 25 & 12 & -2 & 0 & 0 & 0 \\ \hline 75 & 6 & 18 & 8 & 1 & 0 \\ \hline 125 & 8 & 4 & 10 & 10 & 1 \\ \hline 175 & 0 & 5 & 2 & 2 & 2 \\ \hline 225 & 0 & 1 & 4 & 1 & 3 \\ \hline \end{array} \][/tex]

### Step 4: Flatten and Clean the Frequency Data
We will flatten the frequency table and align it with the corresponding midpoints. We will also remove negative and zero frequencies.

Valid frequencies and their corresponding SR and AE:

| Sales Revenue | Ad Expenditure | Frequency |
|---------------|----------------|-----------|
| 25 | 20 | 12 |
| 75 | 20 | 6 |
| 75 | 60 | 18 |
| 75 | 100 | 8 |
| 75 | 140 | 1 |
| 125 | 20 | 8 |
| 125 | 60 | 4 |
| 125 | 100 | 10 |
| 125 | 140 | 10 |
| 125 | 180 | 1 |
| 175 | 60 | 5 |
| 175 | 100 | 2 |
| 175 | 140 | 2 |
| 175 | 180 | 2 |
| 225 | 60 | 1 |
| 225 | 100 | 4 |
| 225 | 140 | 1 |
| 225 | 180 | 3 |

### Step 5: Calculate the Means
Calculate the weighted means of sales revenue and advertisement expenditure.

[tex]\[ \text{Mean Sales Revenue} = \frac{\sum \text{(Frequency} \times \text{Sales Revenue})}{\sum \text{Frequency}} \][/tex]

[tex]\[ \text{Mean Advertisement Expenditure} = \frac{\sum \text{(Frequency} \times \text{Ad Expenditure})}{\sum \text{Frequency}} \][/tex]

Total frequency: [tex]\(12 + 6 + 18 + 8 + 1 + 8 + 4 + 10 + 10 + 1 + 5 + 2 + 2 + 2 + 1 + 4 + 1 + 3 = 98\)[/tex]

Sum Product for SR:
[tex]\[ 12 \times 25 + 6 \times 75 + 18 \times 75 + 8 \times 75 + 1 \times 75 + 8 \times 125 + 4 \times 125 + 10 \times 125 + 10 \times 125 + 1 \times 125 + 5 \times 175 + 2 \times 175 + 2 \times 175 + 2 \times 175 + 1 \times 225 + 4 \times 225 + 1 \times 225 + 3 \times 225 = 10650 \][/tex]

Sum Product for AE:
[tex]\[ 12 \times 20 + 6 \times 20 + 18 \times 60 + 8 \times 100 + 1 \times 140 + 8 \times 20 + 4 \times 60 + 10 \times 100 + 10 \times 140 + 1 \times 180 + 5 \times 60 + 2 \times 100 + 2 \times 140 + 2 \times 180 + 1 \times 60 + 4 \times 100 + 1 \times 140 + 3 \times 180 = 6830 \][/tex]

Mean SR:
[tex]\[ \text{Mean SR} = \frac{10650}{98} \approx 108.68 \][/tex]

Mean AE:
[tex]\[ \text{Mean AE} = \frac{6830}{98} \approx 69.69 \][/tex]

[tex]\[ \text{(Continued in the next response due to length limitations)} \][/tex]

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