A component is classed as defective if it has a diameter of less than 80mm. In a batch of 400 components, the mean diameter is 75mm and the standard deviation is 2.8mm. Assuming the diameters are normally distributed, determine how many are likely to be classed as defective.

To solve this problem, we can use the concept of the standard normal distribution.

Given:
Sample size (n) = 400
Mean (μ) = 75 mm
Standard deviation (σ) = 2.8 mm

We are trying to find the number of components that are likely to be classed as defective, which means the diameter is less than 80 mm.

Step 1: Convert the given values to z-scores using the formula:

$$z = \frac{x - \mu}{\sigma}$$

where z is the z-score, x is the value, μ is the mean, and σ is the standard deviation.

For a diameter of 80 mm:
$$z = \frac{80 - 75}{2.8} \approx 1.79$$

Step 2: Calculate the probability using the standard normal distribution table or a calculator. The probability of a component being defective is the probability of having a diameter less than 80 mm, which is the same as finding the probability of having a z-score less than 1.79.

Step 3: Look up the z-score 1.79 in the standard normal distribution table or use a calculator to find the corresponding probability. From the table, we find that the probability is approximately 0.9633.

Step 4: Calculate the number of components likely to be classed as defective by multiplying the probability by the sample size:

Number of defective components = Probability × Sample size
= 0.9633 × 400
= 385.32

Round the result to the nearest whole number:

Number of defective components ≈ 385

Answer: Based on the given data, it is likely that approximately 385 components will be classed as defective.