Question 1198232: It takes an average of 40 seconds to download a certain file, with a standard deviation of 5 seconds. The actual distribution of the download time is unknown. Using Chebyshev’s inequality, what can be said about the probability of spending more than 1 minute for this download?
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **1. Define the Problem**
* We want to find the probability of spending more than 1 minute (60 seconds) to download the file.
* Mean download time (μ) = 40 seconds
* Standard deviation (σ) = 5 seconds
**2. Calculate the Number of Standard Deviations**
* Difference from the mean: 60 seconds - 40 seconds = 20 seconds
* Number of standard deviations (k): 20 seconds / 5 seconds = 4
**3. Apply Chebyshev's Inequality**
* Chebyshev's Inequality states that for any dataset, the proportion of data that lies within 'k' standard deviations of the mean is at least 1 - (1/k²).
* In this case, k = 4.
* Probability of data within 4 standard deviations of the mean:
* 1 - (1/4²) = 1 - (1/16) = 15/16 = 0.9375
* This means at least 93.75% of the download times will fall within 4 standard deviations of the mean (between 0 and 80 seconds).
**4. Determine the Probability of Spending More Than 1 Minute**
* Since we want the probability of spending *more* than 1 minute (60 seconds), we are looking at the probability of values outside of 4 standard deviations from the mean.
* Probability of spending more than 1 minute:
* 1 - Probability of spending within 4 standard deviations
* 1 - 0.9375 = 0.0625
**Therefore, according to Chebyshev's Inequality, the probability of spending more than 1 minute to download the file is at most 0.0625 or 6.25%.**
**Important Note:**
* Chebyshev's Inequality provides an upper bound on the probability. The actual probability could be lower.
* If the distribution of download times were known (e.g., normal distribution), we could use more precise methods (like the z-score table) to calculate the exact probability.
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