A manufacturer uses a machine to make metal rods - Edexcel - A-Level Maths Statistics - Question 2 - 2022 - Paper 1

To calculate the proportion of metal rods between these lengths, we need to find:

$P(7.94 < L < 8.09)$

Calculating the z-scores for these lengths:

1. For L = 7.94: $z_{7.94} = \frac{7.94 - 8}{0.05} = -1.2$
2. For L = 8.09: $z_{8.09} = \frac{8.09 - 8}{0.05} = 1.8$

Using the z-table, we find the probabilities:

$P(Z < -1.2) \approx 0.1151$ $P(Z < 1.8) \approx 0.9641$

Thus, the proportion is:

$P(7.94 < L < 8.09) = P(Z < 1.8) - P(Z < -1.2) \approx 0.9641 - 0.1151 = 0.849$

First, we need to determine the potential profit from each range of metal rod lengths:

1. For rods less than 7.94 cm:

   • Probability = 0.1151
   • Profit = 0.05 - 0.20 = -0.15

2. For rods between 7.94 cm and 8.09 cm:

   • Probability = 0.849
   • Profit = 0.50 - 0.20 = 0.30

3. For rods longer than 8.09 cm:

   • Probability = 0.0359
   • Profit after shortening = 0.50 - 0.20 - 0.10 = 0.20

Calculating profit per 500 rods:

• Expected profit contribution from fewer than 7.94 cm: $500 \times (0.1151 \times -0.15) = -8.63$
• Expected profit contribution from between 7.94 cm and 8.09 cm: $500 \times (0.849 \times 0.30) = 127.35$
• Expected profit contribution from more than 8.09 cm: $500 \times (0.0359 \times 0.20) = 3.59$

Adding these contributions gives:

$Expected Profit = -8.63 + 127.35 + 3.59 = 122.31.$

Thus, the expected profit for 500 rods is approximately £122.

The manufacturer's aim is for 95% of batches to be acceptable, which translates to a maximum of 6 faulty hinges in a sample of 200.

The probability of a hinge being faulty is 0.015. The expected number of faults in the sample can be calculated using:

$E(X) = n \times p = 200 \times 0.015 = 3$

Using the binomial probability formula, we can assess the likelihood of having fewer than 6 faulty hinges (i.e., $P(X < 6)$). Calculating $P(X \leq 5)$, we find:

$P(X \leq 5) \approx 0.9176$

This suggests that there is a 91.76% chance of a batch being acceptable. Since this is less than the manufacturer's aim of 95%, it is unlikely that they will consistently achieve their goal.