Question

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and standard deviation of 1.2.

i) If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins is at least 51?
ii) without assuming population normality, what is the (approximate) probability that the sample mean hardness for a random sample of 40 pins is at least 51?
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Answer 1

The probability that the sample mean hardness for a random sample of 9 pins is at least 51 is approximately 6.68%. Without assuming population normality, the probability that the sample mean hardness for a random sample of 40 pins is at least 51 is approximately 0.35%.

Step-by-step explanation:

i) To find the probability that the sample mean hardness for a random sample of 9 pins is at least 51, we need to calculate the z-score and then find the area to the right of the z-score on the standard normal distribution curve. The formula for calculating the z-score is: z = (x - μ) / (σ / sqrt(n)), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, x = 51, μ = 50, σ = 1.2, and n = 9. Plugging in the values, we get: z = (51 - 50) / (1.2 / sqrt(9)) = 1.5. Using a standard normal distribution table or a calculator, we can find that the area to the right of a z-score of 1.5 is approximately 0.0668. Therefore, the probability that the sample mean hardness for a random sample of 9 pins is at least 51 is approximately 0.0668 or 6.68%.

ii) Without assuming population normality, we can use the Central Limit Theorem to approximate the distribution of the sample mean. According to the Central Limit Theorem, for a sufficiently large sample size (typically considered to be greater than or equal to 30), the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution. In this case, the sample size is 40, which is considered large enough.

Therefore, we can use the standard normal distribution to find the probability. Following the same steps as in part i), we calculate the z-score using the formula: z = (x - μ) / (σ / sqrt(n)). Plugging in the values, we get: z = (51 - 50) / (1.2 / sqrt(40)) = 2.683. Using a standard normal distribution table or a calculator, we can find that the area to the right of a z-score of 2.683 is approximately 0.0035. Therefore, the probability that the sample mean hardness for a random sample of 40 pins is at least 51 is approximately 0.0035 or 0.35%.