Final answer:
To find the mean diameter and standard deviation for the sample, we can use the formulas for mean and standard deviation. The mean diameter for the sample is 201 inches and the standard deviation is 0.494 inches. The probability that the mean diameter of the sample shafts would differ from the population mean by less than 0.2 inches is 1, indicating that the company's diameter claim is plausible.
Step-by-step explanation:
To find the mean diameter and standard deviation for the sample, we can use the formulas for mean and standard deviation.
The mean of a sample is equal to the mean of the population. Therefore, the mean diameter for the sample is 201 inches.
The standard deviation for the sample can be calculated using the formula: standard deviation = square root of (variance / sample size).
In this case, the variance is 7.84 and the sample size is 89. Plugging in these values, we get:
standard deviation = square root of (7.84 / 89) = 0.494 inches.
To find the probability that 50 randomly selected screws will be within the stated tolerance levels, we need to determine the z-score for the given tolerance level of 0.2 inches.
The formula to calculate the z-score is: z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values, we get:
z = (0.2 - 0) / (0.494 / √50) = 4.05.
Using a standard normal distribution table, we can find that the probability of a z-score less than 4.05 is approximately 1.
Therefore, the probability that the mean diameter of the sample shafts would differ from the population mean by less than 0.2 inches is 1.
Since the probability is 1, it is extremely likely that 50 randomly selected screws will be within the stated tolerance levels. Therefore, the company's diameter claim is plausible.