A factory produces jars of jam and jars of marmalade - AQA - A-Level Maths Mechanics - Question 18 - 2021 - Paper 3

To find the probability that the weight of jam in a jar is greater than 368 grams, we first need to calculate the Z-score for 368 grams. The formula for Z-score is:

Z = rac{X - ext{mean}}{ ext{standard deviation}}

Substituting the values:

= -1.142857$$ We then look up this Z-score in the standard normal distribution table or use a calculator to find: $$P(X > 368) = 1 - P(Z < -1.142857) \ = 1 - 0.1269 \ herefore P(X > 368) = 0.8731$$

Using the inverse normal distribution, for the area of 0.975, we find the corresponding Z-value is approximately 1.96. Therefore,

$P(Z < 1.96) = 0.975$

Given the transformation from normal distribution, we have:

$346 - \mu = 1.96\sigma$

Thus, the equation holds true if $heta$ represents $\\mu$.

From the normal distribution table, we find that the Z-value corresponding to $P(Y < 336) = 0.14$ is approximately -1.08. This gives us:

$336 - \mu = -1.08\sigma$

Now we have two equations: 1) $346 - \mu = 1.96\sigma$ 2) $336 - \mu = -1.08\sigma$

Rearranging both equations allows us to isolate $\\mu$ and $\\sigma$. Solving this system of equations, we find:

From equation 1: \ $\mu = 346 - 1.96\sigma$ Substituting into equation 2 yields: $336 - (346 - 1.96\sigma) = -1.08\sigma$ This simplifies to: $-10 = -1.08\sigma + 1.96\sigma$

\\sigma = -\frac{10}{0.88} = 11.36$$ Substituting back to find $\\mu$ gives: $$\mu = 346 - 1.96(11.36) = 346 - 22.3086 = 323.6914$$