A manufacturer uses a machine to make metal rods. The length of a metal rod, L cm, is normally distributed with - a mean of 8 cm - a standard deviation of x cm Given that the proportion of metal rods less than 7.902 cm in length is 2.5% (a) show that x = 0.05 to 2 decimal places. (b) Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length. The cost of producing a single metal rod is 20p A metal rod - where L 8.09 is shortened for an extra cost of 10p and then sold for 50p (c) Calculate the expected profit per 500 of the metal rods. Give your answer to the nearest pound. The same manufacturer makes metal hinges in large batches. The hinges each have a probability of 0.015 of having a fault. A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty. The manufacturer's aim is for 95% of batches to be accepted. (d) Explain whether the manufacturer is likely to achieve its aim.

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

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A manufacturer uses a machine to make metal rods. The length of a metal rod, L cm, is normally distributed with - a mean of 8 cm - a standard deviation of x cm Giv... show full transcript

Worked Solution & Example Answer:A manufacturer uses a machine to make metal rods - Edexcel - A-Level Maths Statistics - Question 2 - 2022 - Paper 1

Step 1

show that x = 0.05 to 2 decimal places.

96%

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Answer

To find the value of x, we use the standard normal distribution. The z-score corresponding to the lower 2.5% is -1.96 (from z-tables). Thus,

P(L < 7.902) = 0.025 = P\left( \frac{L - 8}{x} < -1.96 \right)

Rearranging gives:

\frac{L - 8}{x} = -1.96 \implies L = -1.96x + 8

Substituting L = 7.902 into the equation, we have:

7.902 = -1.96x + 8

Solving this gives:

1.96x = 8 - 7.902 = 0.098 \ \Rightarrow x = \frac{0.098}{1.96} = 0.05

Thus, it is shown that x = 0.05.

Step 2

Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length.

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Answer

To calculate the proportion of metal rods between 7.94 cm and 8.09 cm, we need the z-scores:

For L = 7.94:

Z = \frac{7.94 - 8}{0.05} = -1.2

For L = 8.09:

Z = \frac{8.09 - 8}{0.05} = 1.8

Using the z-table or calculator, we find:

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

Thus, the proportion:

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

Step 3

Calculate the expected profit per 500 of the metal rods.

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Answer

To calculate the expected profit for 500 rods, we need to find the income from each category:

For rods less than 7.94 cm:
- Probability = 0.30
- Selling price = 0.05
- Profit = 0.05 - 0.20 = -0.15
For rods between 7.94 cm and 8.09 cm:
- Probability = 0.20
- Selling price = 0.50
- Profit = 0.50 - 0.20 = 0.30
For rods longer than 8.09 cm:
- Probability = 0.50
- Shortening cost = 0.10
- Selling price = 0.50
- Profit = 0.50 - (0.20 + 0.10) = 0.20

Calculating expected profit:

E(Profit) = (0.30 \times -0.15) + (0.20 \times 0.30) + (0.50 \times 0.20)

= -0.045 + 0.06 + 0.10 = 0.115

For 500 rods, the expected profit:

Expected Profit = 500 \times 0.115 = 57.5 p \\ ext{(to the nearest pound: } £1 ext{)}

Step 4

Explain whether the manufacturer is likely to achieve its aim.

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Answer

Let X be the number of faulty hinges in a random sample of 200. The probability of a hinge being faulty, p = 0.015. Thus, np = 200 x 0.015 = 3 faulty hinges.

Using the binomial distribution, we can calculate:

$P(X < 6) = 1 - P(X \geq 6)$

Using a normal approximation: Mean: μ = np = 3; Variance: σ² = np(1 - p) = 2.955. Standard deviation: σ = 1.721.

To find the probability: Convert to a z-score:

P(X \geq 6) = P\left(Z \geq \frac{6 - 3}{1.721} \right)$$ Calculating this results in a z-score of approximately 1.742, which corresponds to a value of about 0.0842. So, $$P(X < 6) = 1 - 0.0842 = 0.9158.

Since 0.9158 < 0.95, the manufacturer is unlikely to achieve its aim.

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

Worked Solution & Example Answer:A manufacturer uses a machine to make metal rods - Edexcel - A-Level Maths Statistics - Question 2 - 2022 - Paper 1

show that x = 0.05 to 2 decimal places.

Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length.

Calculate the expected profit per 500 of the metal rods.

Explain whether the manufacturer is likely to achieve its aim.

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