Final answer:
To find the probability of the temperature being within a specific range in a kiln, calculate the z-scores for the range limits and look up the corresponding probabilities in the standard normal distribution table. Subtract the smaller probability from the larger to find the probability of the temperature being within that range. Approximately 68% of the data falls within one standard deviation from the mean in a normal distribution.
Step-by-step explanation:
To calculate the probability that the temperature at a random location in the kiln falls within a specific range, we use the properties of the normal distribution. Since the mean temperature (µ) in the kiln is 900 degrees and the standard deviation (σ) is 60 degrees, we can apply the standard normal distribution (z-score) formula to find probabilities for specific ranges.
For instance, if we are interested in finding the probability of the temperature being between 840 degrees and 960 degrees, we'd first calculate the z-scores for both temperatures:
- Z1 for 840 degrees = (840 - 900) / 60 = -1
- Z2 for 960 degrees = (960 - 900) / 60 = 1
Next, we'd consult the standard normal distribution table (or use a calculator) to find the probabilities for these z-scores and then subtract the smaller z-score's probability from the larger one.
The range provided in this example, 840 to 960 degrees, corresponds to the temperatures within one standard deviation from the mean. The empirical rule tells us that approximately 68% of the data in a normal distribution falls within one standard deviation of the mean, making this a probability representation of the temperatures we can expect in that range.
If the range is different, the steps are the same, but you would use the actual values to compute the respective z-scores.