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

To find this probability, we will use the normal distribution properties:

1. First, calculate the z-score for $X = 368$: $z = \frac{X - \mu}{\sigma} = \frac{368 - 372}{3.5} = \frac{-4}{3.5} \approx -1.14$

2. Next, find the probability: $P(X > 368) = 1 - P(X < 368)$ From z-tables or standard normal distribution calculators, we find: $P(X < 368) \approx 0.1271$

3. Finally, calculate: $P(X > 368) = 1 - 0.1271 = 0.8729$

To show this, we start by recognizing the standard normal model:

1. Using the inverse of the standard normal distribution, we know: $z_{0.975} \approx 1.96$

2. The z-score is calculated as: $z = \frac{346 - \mu}{\sigma}$

3. Setting this equal to 1.96: $\frac{346 - \mu}{\sigma} = 1.96$

4. Rearranging gives: $346 - \mu = 1.96 \sigma$

Hence, we have established the required equation.

1. From the normal distribution, we find the z-score associated with $P(Y < 336) = 0.14$: $z_{0.14} \approx -1.08$

2. This gives us: $\frac{336 - \mu}{\sigma} = -1.08$

3. Rearranging yields: $336 - \mu = -1.08 \sigma$ [Equation 1]

4. We now have a system of equations:

   • Equation 1: $336 - \mu = -1.08 \sigma$
   • Equation 2: $346 - \mu = 1.96 \sigma$

5. Solving these equations:

   • From Equation 1, we can express $\\mu$: $\mu = 336 + 1.08 \sigma$

6. Substituting this into Equation 2: $346 - (336 + 1.08 \sigma) = 1.96 \sigma$

   • Simplifying: $10 = 1.96 \sigma + 1.08 \sigma$ $10 = 3.04 \sigma$ $\sigma \approx 3.29$

7. Finally, substituting back to find $μ$:

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u = 341.4$$