Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)
Mean for x: 10.5
Mean for y: 8.9
What is the lower limit for x?
What is the upper limit for x
What is the lower limit for y?
What is the upper limit for y?
The Correct Answer and Explanation is:
To compute a 75% Chebyshev interval around the means of the given data points, we can use Chebyshev’s inequality, which states that for any ( k > 1 ), at least ( 1 – \frac{1}{k^2} ) of the data falls within ( k ) standard deviations of the mean. For a 75% interval, we can determine ( k ) as follows:
[
1 – \frac{1}{k^2} = 0.75
]
[
\frac{1}{k^2} = 0.25 \implies k^2 = 4 \implies k = 2
]
This means that at least 75% of the data falls within 2 standard deviations of the mean.
Step-by-Step Calculation:
- Determine the Mean Values:
- Mean for ( x ) (( \mu_x )) = 10.5
- Mean for ( y ) (( \mu_y )) = 8.9
- Standard Deviations:
- For this problem, we will need the standard deviations (( \sigma_x ) and ( \sigma_y )). However, since they are not provided, let’s denote the standard deviations as ( \sigma_x ) and ( \sigma_y ).
- Calculate the Chebyshev Interval:
- The lower and upper limits for ( x ) and ( y ) can be calculated using the formulas:
- Lower limit for ( x ) = ( \mu_x – k \cdot \sigma_x )
- Upper limit for ( x ) = ( \mu_x + k \cdot \sigma_x )
- Lower limit for ( y ) = ( \mu_y – k \cdot \sigma_y )
- Upper limit for ( y ) = ( \mu_y + k \cdot \sigma_y )
Given that ( k = 2 ):
Formulas for the Limits:
- For ( x ):
- Lower limit for ( x ) = ( 10.5 – 2\sigma_x )
- Upper limit for ( x ) = ( 10.5 + 2\sigma_x )
- For ( y ):
- Lower limit for ( y ) = ( 8.9 – 2\sigma_y )
- Upper limit for ( y ) = ( 8.9 + 2\sigma_y )
Final Calculation (with Sample Standard Deviations):
If we had specific values for ( \sigma_x ) and ( \sigma_y ), we could compute exact limits. For example, if we assume:
- ( \sigma_x = 1.5 ) and ( \sigma_y = 2.0 ):
- For ( x ):
- Lower limit for ( x ) = ( 10.5 – 2(1.5) = 10.5 – 3 = 7.5 )
- Upper limit for ( x ) = ( 10.5 + 2(1.5) = 10.5 + 3 = 13.5 )
- For ( y ):
- Lower limit for ( y ) = ( 8.9 – 2(2.0) = 8.9 – 4 = 4.9 )
- Upper limit for ( y ) = ( 8.9 + 2(2.0) = 8.9 + 4 = 12.9 )
Conclusion
The lower and upper limits based on our assumed standard deviations would be:
- Lower limit for ( x ): 7.50
- Upper limit for ( x ): 13.50
- Lower limit for ( y ): 4.90
- Upper limit for ( y ): 12.90
Thus, we conclude that Chebyshev’s inequality provides a reliable way to estimate the spread of data around the mean, accommodating various distributions.