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

To find the proportion of metal rods between 7.94 cm and 8.09 cm, we again use the standard normal distribution. First, we find the z-scores for both lengths:

For $L = 7.94$:

z_1 = rac{7.94 - 8}{x} = rac{7.94 - 8}{0.05} = -1.20

For $L = 8.09$:

z_2 = rac{8.09 - 8}{x} = rac{8.09 - 8}{0.05} = 1.80

Now we consult the standard normal distribution table to find the probabilities:

To calculate the expected profit per 500 metal rods, we first determine the income from selling each category:

1. For $L < 7.94$:
   • Proportion: $0.025$, Profit: $0.05 - 0.20 = -0.15$ (Loss)
2. For $7.94 ext{ to } 8.09$:
   • Proportion: $0.849$, Profit: $0.50 - 0.20 = 0.30$
3. For $L > 8.09$:
   • Proportion: $0.126$, Profit: $(0.50 - 0.20 - 0.10) = 0.20$

The expected profit for 500 rods is thus: $ext{Expected profit} = 500 imes (0.025 imes -0.15 + 0.849 imes 0.30 + 0.126 imes 0.20)$ Calculating this gives:

The final expected profit is approximately £122.

To determine if the manufacturer is likely to achieve the aim of 95% acceptance of hinges, we can use the binomial distribution as the probability of having a faulty hinge is given:

The probability of a hinge being faulty is $p = 0.015$. We want to find the probability of getting more than 6 faulty hinges out of 200: By using the binomial cumulative distribution:

$P(X ext{ allows acceptance}) = P(X < 6)$ Using a binomial table or calculator: $P(X < 6) = 0.9176$

This indicates that the likelihood of acceptance is about 91.76%, which is less than the desired 95%. Therefore, the manufacturer is unlikely to achieve their aim.