The rechargeable battery made by company A for E-bike had an average durability of 240 hours and a standard deviation of 20 hours after being fully charged. If a sample of 6 fully charged batteries is taken, find the probability that the sample mean exceeds 235 hours.a.0.2500b.0.7291c.1.000d.0.2709
Question
The rechargeable battery made by company A for E-bike had an average durability of 240 hours and a standard deviation of 20 hours after being fully charged. If a sample of 6 fully charged batteries is taken, find the probability that the sample mean exceeds 235 hours.a.0.2500b.0.7291c.1.000d.0.2709
Solution
To solve this problem, we need to use the concept of sampling distribution of the mean. Here are the steps:
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Identify the given values:
- Population mean (μ) = 240 hours
- Population standard deviation (σ) = 20 hours
- Sample size (n) = 6
- Sample mean (x̄) = 235 hours
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Calculate the standard error of the mean (SE). The standard error is calculated as the standard deviation divided by the square root of the sample size: SE = σ / √n = 20 / √6 ≈ 8.16
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Calculate the z-score. The z-score is the number of standard errors that x̄ is away from μ. It is calculated as: Z = (x̄ - μ) / SE = (235 - 240) / 8.16 ≈ -0.61
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Look up the z-score in the z-table to find the probability. The z-table tells us the probability that the sample mean is less than 235 hours. But we want the probability that the sample mean is greater than 235 hours. So, we subtract the value from the z-table from 1.
The value from the z-table for -0.61 is 0.2709. So, the probability that the sample mean exceeds 235 hours is 1 - 0.2709 = 0.7291.
So, the correct answer is b. 0.7291.
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