The number of days (X) after mailout it takes a utility company to receive payment for a customer’s bill is a normal random variable with a mean of 31 days and a standard deviation of 6 days.The probability is 0.997 that a bill will be received between which of the following number of days? (13, 49) (19, 43) (25, 37)
Question
The number of days (X) after mailout it takes a utility company to receive payment for a customer’s bill is a normal random variable with a mean of 31 days and a standard deviation of 6 days.The probability is 0.997 that a bill will be received between which of the following number of days? (13, 49) (19, 43) (25, 37)
Solution
To solve this problem, we need to use the properties of a normal distribution.
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First, we need to understand that the probability of 0.997 corresponds to a z-score of approximately ±3 (from the standard normal distribution table or using a z-score calculator).
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The z-score is calculated using the formula: Z = (X - μ) / σ, where X is the value from the data set, μ is the mean, and σ is the standard deviation.
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We can rearrange the formula to solve for X: X = Z*σ + μ.
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To find the range of days, we need to calculate X for both Z = -3 and Z = 3.
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For Z = -3: X = -3*6 + 31 = 13.
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For Z = 3: X = 3*6 + 31 = 49.
So, the probability is 0.997 that a bill will be received between 13 and 49 days.
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