Final answer:
To calculate the probability that the average strength of 55 steel beams is between 17 and 18 kilonewtons, we determine the standard error, convert the range to z-scores, and use these scores to find the cumulative probabilities from the standard normal distribution.
Step-by-step explanation:
To find the probability that the average strength of these 55 beams is between 17 and 18 kilonewtons (kn), we need to use the Central Limit Theorem which implies that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough. Given the average strength (μm) is 17.2 kn and the standard deviation (σ) is 2.5 kn for the general population of beams, we can compute the standard error of the mean (SEM) for a sample of 55 beams using the formula SEM = σ/√(n), where n is the sample size.
Here, SEM = 2.5 kn / √(55) = 2.5 kn / 7.42 = 0.337 kn. Now, we convert the desired range of 17 kn to 18 kn to a z-score. The z-score for a value x is given by z = (x - μm)/SEM. For 17 kn, z = (17 - 17.2) / 0.337 = -0.594. For 18 kn, z = (18 - 17.2) / 0.337 = 2.374. We can now use a z-table or standard normal distribution calculator to find the probabilities corresponding to these z-scores.
The probability that a value is between z = -0.594 and z = 2.374 is the difference between the probabilities at z = 2.374 and z = -0.594. Assuming a standard normal distribution, these probabilities can be looked up in a z-table or calculated using appropriate statistical software. The result will give us the probability that the average strength of the sample of 55 steel beams is between 17 and 18 kn.