Toothpaste Market Research Survey Analysis

Since the price interval > 10 is open-ended, the exact midpoint for this highest category cannot be specified without upper-bound information. Consequently, the precise mean price cannot be determined.

Total households with income $\le 5,000$: Col 1 ($< 1500$) = $1500 + 1750 + 500 + 250 + 50 = 4050$ Col 2 ($1500\text{--}2500$) = $500 + 750 + 1200 + 450 + 100 = 3000$ Col 3 ($2500\text{--}3500$) = $350 + 400 + 500 + 250 = 1500$ Col 4 ($3500\text{--}5000$) = $150 + 200 + 400 + 300 = 1050$ Total = $4050 + 3000 + 1500 + 1050 = 9600$.

Households among these using toothpaste $< 8$ (rows $<4$, $4\text{--}6$, $6\text{--}8$): Row $< 4$: $1500 + 500 = 2000$ Row $4\text{--}6$: $1750 + 750 + 350 + 150 = 3000$ Row $6\text{--}8$: $500 + 1200 + 400 + 200 = 2300$ Total = $2000 + 3000 + 2300 = 7300$.

Probability = $\frac{7300}{9600} \approx 0.7604 \approx 0.76$.

Consider the row for price between Rs. 6 and Rs. 8: Total households in this row = $500 + 1200 + 400 + 200 + 50 = 2350$.

Households in this row with income $\ge 2,500$ (columns $2500\text{--}3500$, $3500\text{--}5000$, $>5000$): $400 + 200 + 50 = 650$.

Probability = $\frac{650}{2350} \approx 0.2766 \approx 0.28$.

Total households in column $< 1500$ = $1500 + 1750 + 500 + 250 + 50 = 4050$.

Sum of all entries across all columns: Col 1 = 4050 Col 2 = 3000 Col 3 = 1500 Col 4 = 1050 Col 5 ($>5000$) = $0 + 50 + 50 + 100 + 200 = 400$ Total households = $4050 + 3000 + 1500 + 1050 + 400 = 10000$.

Ratio = $\frac{4050}{10000} = 0.405$.

Let's evaluate each statement:

• Statement I: For low-priced toothpastes ($< 4$ and $4\text{--}6$), household counts decrease steadily as income increases. True.
• Statement II: For income $< 1500$, there are still 50 households using $>10$ priced toothpastes, so this statement is false.
• Statement III: Households with income $< 2500$ = $4050 + 3000 = 7050$, which is $70.5\%$ of $10000$, well over $50\%$. True.

Thus, statements I and III are definitely true.