Keep the instructed rounded model consistent with later probabilities

Let $X\sim N(\mu,15^2)$ and $Y\sim N(450,20^2)$ be masses in grams. $P(X>300)=0.65$ gives $P(X\le300)=0.35$, so $P(Z\le(300-\mu)/15)=0.35$. Hence $(300-\mu)/15=-0.38532$ and μ≈305.78, giving 306 to the nearest whole number. The source’s standardisation line uses X in place of Z, and labels 305.78 “3 s.f.”;305.78 actually has 5 significant figures. The required whole-number result is 306.

Using the instructed $X\sim N(306,15^2)$ and independently selected fruit, $S=X_1+X_2+X_3\sim N(3(306),3(15^2))=N(918,675)$. Thus $P(S<950)=0.89097\approx0.891$ to 3 significant figures. The source labels this event P(X<950) after defining the sum; S here avoids that notational ambiguity.

The source uses $W\sim B(10,0.65)$ and $P(W>5)=1-P(W\le5)=0.751496\approx0.751$. This retains the original 65% probability. Under the explicit instruction to take μ=306, however, $p=P(X>300)=\Phi(0.4)\approx0.655421742$, so $W\sim B(10,p)$ gives $P(W>5)\approx0.763267838\approx0.763$. The two inputs are inconsistent after rounding the mean; the source 0.751 is labelled rather than silently substituted for the instructed-model result.

$D=X_1+X_2+X_3-(Y_1+Y_2)\sim N(18,1475)$. The parameters come from mean 918−900 and variance 675+800. Therefore $P(|D|\le100)=P(-100\le D\le100)=0.98256\approx0.983$ to 3 significant figures. Absolute-value bars are retained from the source image.

$T=0.007(X_1+X_2+X_3+X_4)\sim N(0.007(4)(306),0.007^2(4)(15^2))=N(8.568,0.0441)$. Thus $P(T\ge9)=0.019836\approx0.0198$ to 3 significant figures.

$S=0.007X+0.008Y\sim N(0.007(306)+0.008(450),0.007^2(15^2)+0.008^2(20^2))=N(5.742,0.036625)$. Following the source, “not making a loss” means the fixed promotional price is at least the ordinary mass-based selling value S; no purchase costs are supplied.

$P(S\le a)\ge0.90$ requires $a\ge5.9873$ dollars. The least cent-denominated price is 5.99 dollars, as in the source. Without restricting prices to cents, the lower bound is the unrounded 90 th percentile.