How to Find Intervals for α and β in a Linear Model Given y and x?

In summary, the interval for coefficients is a range of values used to determine the uncertainty associated with the estimated value of a coefficient. It is calculated using statistical methods and can be affected by factors such as sample size, confidence level, and data variability. The interval can be negative or zero, and this information can be used in data analysis to assess the significance of a coefficient or compare different models.
  • #1
zli034
107
0
For a simple linear model:
[itex]\alpha+\beta\times x=y[/itex]

If it is observed that [itex]y \in (-8.51,23.20) [/itex] given [itex]x=4[/itex]

The question is to give intervals of [itex]\alpha, \beta[/itex], which satisfy [itex]y \in (-8.51,23.20) [/itex] given [itex]x=4[/itex].

Is this problem identifiable? Can it be found the unique intervals for [itex]\alpha, \beta[/itex]?

I am guessing this is not a well defined problem. How to improve on this?
 
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  • #2
α and β are connected by a linear equation. This means the you can give any value for one variable and get an interval for the other. To improve you need another relation.
 

FAQ: How to Find Intervals for α and β in a Linear Model Given y and x?

What is the purpose of finding the interval for coefficients?

The interval for coefficients is used to determine the range in which the true value of a coefficient lies. This helps in understanding the level of uncertainty associated with the estimated value of the coefficient.

2. How is the interval for coefficients calculated?

The interval for coefficients is calculated using statistical methods, such as confidence intervals or standard errors, which take into account the variability in the data and the sample size. These methods provide a range of values within which the true value of the coefficient is likely to fall.

3. What factors can affect the interval for coefficients?

The main factors that can affect the interval for coefficients are the sample size, the level of confidence chosen, and the variability in the data. A larger sample size, a higher confidence level, and lower variability will result in a narrower interval for coefficients.

4. Can the interval for coefficients be negative or zero?

Yes, the interval for coefficients can be negative or zero. This means that the true value of the coefficient can be negative or zero with a certain level of confidence. It is important to interpret the interval in the context of the data and the research question being studied.

5. How can the interval for coefficients be used in data analysis?

The interval for coefficients can be used to assess the significance of a particular coefficient in a statistical model. If the interval does not include zero, then the coefficient is considered to be statistically significant. It can also be used to compare different models and determine which one has a more precise estimation of the coefficients.

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