Chebyshev's Theorem Limit Question-Value Greater than 1

[reemsh121]

Homework Statement

According to Nielsen Media Research, kids ages 12-17 watch an average of 3 hours of tv per day. Suppose that the standard deviation is 1 hour and that the distribution of time spent watching tv has a bell-shaped distribution.

a) what percentage of kids aged 12-17 watch tv between 2 and 3 hours per day?

Homework Equations

1-1/z^2

The Attempt at a Solution

The standard deviation of 2 hours is 1, from the mean. But the value of z has to be greater than 1. With a question that has a range in it, usually means both sides will have a standard deviation from the mean. But one side is 1 and the other is the same as the mean.

So we only have one side, and I divided 1 in half to get .5 and plugged it into the formula.

1-1/.5^2

1-1/.25

1-4

-3

The answer is 34%.

I know I'm missing a step but I don't know what step. The formula is pretty straight forward and I get it, but this problem is really messing me up.

 

[diazona]

Chebyshev's inequality only makes a general statement about any probability distribution. So for example, if you plug in z = 1, you get 1-1/z^2 = 0, which means that no less than 0 of the values are more than 1 standard deviation away from the mean. But in this case, you know that the distribution is normal (a bell curve), so you can get a more precise answer, which is going to be the 34%.

As for how to get that 34%... look it up maybe? Or if you know the formula for the cumulative normal distribution (the error function), you can use that.

 

[Architeuthis Dux]

I would like to clarify a few things about the problem and the attempted solution. Firstly, Chebyshev's theorem is used to calculate the percentage of data that falls within a certain number of standard deviations from the mean, not within a specific range. Secondly, the formula for Chebyshev's theorem is (1 - 1/z^2), where z is the number of standard deviations from the mean. In this case, z = 1, since we are looking at one standard deviation from the mean.

To answer the question, we can use the empirical rule, which states that approximately 68% of the data falls within one standard deviation from the mean, in a bell-shaped distribution. Therefore, approximately 68% of kids aged 12-17 watch tv between 2 and 3 hours per day. This aligns with the answer of 34% obtained in the attempted solution, as 34% is half of 68%.

In summary, Chebyshev's theorem is not applicable in this situation and the empirical rule should be used instead. It is important to understand the assumptions and limitations of different statistical tools and use them appropriately.