Final answer:
To calculate the probability of demand for a product falling within specific ranges, the demand is standardized to z-scores and the probabilities are derived using a z-table. Probabilities for less than 5000 and greater than 8000 are calculated directly, while the probability for the range between 5000 and 8000 is found by subtracting the two probabilities.
Step-by-step explanation:
Probability Calculation Using Normal Distribution
To calculate the probability that demand for a product is within certain ranges, we first need to standardize the demand values to z-scores. This involves subtracting the mean from the demand value and dividing by the standard deviation. With a mean (μ) of 6000 and a standard deviation (σ) of 1200 for the demand, the z-score formula is as follows:
z = (X - μ) / σ
For the demand, the calculations are:
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Using a z-table, find the area to the left of the z-score for part (a) and (b), and the area to the right of the z-score for part (c). For a z-score of -0.8333 (a), the probability is approximately 0.2023. For a z-score of 1.6667 (c), the probability is 1 - 0.9525 = 0.0475 (since the table gives the area to the left). For part (b), subtract the probability of being less than 5000 from the probability of being less than 8000 (calculate the z-score and look it up in the table).
Remember, the total area under the normal curve represents a probability of 1, so subtracting the area to the left from 1 gives the area to the right.