a) Calculate the coefficient of correlation between the price and sales. [4] Price 25 21 28 26 20 30 Sales 60 54 66 68 70 50 b) The chance that A can solve the problem is 3/5 and the chance that B…

a) Coefficient of Correlation between Price and Sales

The coefficient of correlation ($r$) between two variables $X$ (Price) and $Y$ (Sales) is calculated using the formula:

$$r=\frac{n\sum XY-(\sum X)(\sum Y)}{\sqrt{[n\sum X^2-(\sum X{)}^2][n\sum Y^2-(\sum Y{)}^2]}}$$

Step 1: Organize the data

We have the following data:

Step 2: Calculate the required sums

$$\sum X=25+21+28+26+20+30=150$$ $$\sum Y=60+54+66+68+70+50=368$$ $$\sum XY=(25\times 60)+(21\times 54)+(28\times 66)+(26\times 68)+(20\times 70)+(30\times 50)$$ $$=1500+1134+1848+1768+1400+1500=8150$$ $$\sum X^2=25^2+21^2+28^2+26^2+20^2+30^2$$ $$=625+441+784+676+400+900=3826$$ $$\sum Y^2=60^2+54^2+66^2+68^2+70^2+50^2$$ $$=3600+2916+4356+4624+4900+2500=22896$$

Step 3: Calculate the numerator and denominator

$$\text{Numerator}=n\sum XY-(\sum X)(\sum Y)=6\times 8150-150\times 368=48900-55200=-6300$$

$$\text{Denominator}=\sqrt{[n\sum X^2-(\sum X{)}^2][n\sum Y^2-(\sum Y{)}^2]}$$ $$=\sqrt{[6\times 3826-150^2][6\times 22896-368^2]}$$ $$=\sqrt{[22956-22500][137376-135424]}$$ $$=\sqrt{[456][1952]}=\sqrt{890352}=943.57$$

Step 4: Calculate the coefficient of correlation

$$r=\frac{-6300}{943.57}=-0.6676$$

Final answer: The coefficient of correlation between price and sales is -0.6676.

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b) Probability of Solving the Problem

Given:

• Probability that A can solve the problem, $P(A)=\frac{3}{5}$
• Probability that B can solve the problem, $P(B)=\frac{2}{3}$

i) Probability that the problem is solved

The problem is solved if either A or B or both solve it. We use the formula for the probability of the union of two events:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Assuming that A and B solve the problem independently, the probability that both solve it is:

$$P(A\cap B)=P(A)\times P(B)=\frac{3}{5}\times \frac{2}{3}=\frac{6}{15}=\frac{2}{5}$$

Now, calculate $P(A\cup B)$:

$$P(A\cup B)=\frac{3}{5}+\frac{2}{3}-\frac{2}{5}$$

Convert to a common denominator (15):

$$P(A\cup B)=\frac{9}{15}+\frac{10}{15}-\frac{6}{15}=\frac{13}{15}$$

Final answer: The probability that the problem is solved is $\frac{13}{15}$.

ii) Probability that none of them can solve the problem

The probability that neither A nor B solves the problem is:

$$P(\text{None})=1-P(A\cup B)=1-\frac{13}{15}=\frac{2}{15}$$

Alternatively, calculate it directly:

$$P(\text{None})=P(\text{Not }A)\times P(\text{Not }B)=\left(1-\frac{3}{5}\right)\times \left(1-\frac{2}{3}\right)=\frac{2}{5}\times \frac{1}{3}=\frac{2}{15}$$

Final answer: The probability that none of them can solve the problem is $\frac{2}{15}$.