Following are the solution to the given points:
Assume A, P, and B is the masses of apples, pears, and basket, respectively.
[tex]\to W = 4A + 6P + B[/tex]
Calculating the mean of W:
[tex]\to \bold{W = E(W) }[/tex]
[tex]\bold{= E(4A + 6P + B)} \\\\ \bold{= 4E(A) + 6E(P) + E(B)}\\\\\bold{= 4 \times 4.72 + 6 \times 5.41 + 13.52} \\\\ \bold{= 18.88 + 32.46 + 13.52} \\\\ \bold{= 64.86 }[/tex]
Calculating the variance of W:
[tex]\to \bold{W=Var(W)}\\\\[/tex]
[tex]\bold{= Var(4A + 6P + B)}\\\\ \bold{= 42 Var(A) + 62 Var(P) + Var(B)}\\\\\bold{= 16 \times 0.22 + 36 \times 0.182 + 1.882}\\\\ \bold{= 3.52 + 6.552 + 1.882}\\\\ \bold{= 5.3408}[/tex]
Calculating the standard deviation of W:
[tex]\to \bold{W= \sqrt{5.3408}= 2.311017}[/tex]
The linear combination of Normal random variables is known to follow the normal distribution.
[tex]\to \bold{W \sim N(64.86 , 2.311017)}[/tex]
Calculating the Probability of the Total Weight is more difficult.[tex]\to \bold{67 = P(W > 67)}[/tex]
[tex]\bold{= P[ Z > \frac{(67 - 64.86)}{ 2.311017}]}\\\\\bold{= P[Z > 0.93]}\\\\\bold{= 0.1762}[/tex] (Using Standard Normal distribution)
Therefore, the final answer is "64.86, 2.311017, and 0.1762".
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