Chebyshev's Theorem: The percentage of observations that are within k standard deviations of the mean is at least
100(1 - (1/k2))%
Chebyshev's Theorem is applicable to ANY data set, whether skewed or symmetrical.
Empirical Rule: For a symmetrical data set, approximately
- 68% of the data is within 1 standard deviation from the mean
- 95% of the data is within 2 standard deviations from the mean
- 99.7% of the data is within 3 standard deviations from the mean
The Attempt at a Solution
I'm a little confused as to how to solve these problems. Based on the wording, I think I'm supposed to be using Chebyshev's Theorem and the Empirical Rule, but I'm not sure.
I solved (a) using Chebyshev's Theorem, since at that point in the assignment we haven't been told whether the data is symmetric (has a bell shape) or not:
mean = 1500
standard deviation = 80
Since the problem is asking for the light bulbs that lasted between 1300 and 1700 hours until failure - that is, the interval (1300, 1700) - I note that for k = 3 standard deviations from the mean, we have an interval of hours of operation until failure of
(mean - 3 (standard deviation), mean + 3 (standard deviation)) = (1500 - 3(80) , 1500 + 3(80)) = (1260, 1740)
which would indicate that the percentage of the light bulbs that lasted between the specified interval is given by
100(1 - 1/(3)2)% = 100(1- (1/9))% ≈ 89%
This is where I'm a little confused. The problem is asking for the number of light bulbs whose hours of operation fall within the interval (1300, 1700), but the interval given by adding/subtracting 3 standard deviations is (1260, 1740), which is a little too far; by contrast, the interval given by using k = 2 isn't far enough.
Now, I have noticed that the interval (1300, 1700) ⊆ (1260, 1740) since every point in the former is a member of the latter, but does that make it okay to apply Chebyshev's Rule here?
I have a similar concern about (b). Since the data set is bell-shaped and thus roughly symmetrical, I used the Empirical Rule.
The interval given by the problem is (1340, 1580).
Since the interval calculated by using 2 standard deviations from the mean is
(1500 - 2(80) , 1500 + 2(80)) = (1340, 1660)
and (1340, 1580) ⊆ (1340, 1660) by the same reasoning used earlier, I conclude that based on the Empirical Rule, at least 95.4% of values must fall within the specified range, which would yield
500 * (0.954) = 477 lightbulbs
lasted between 1340 and 1580 hours. Is this correct, or am I misusing these rules? I've seen some classmates doing calculations that involved p-values and z-scores, am I supposed to be using those instead?
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