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Discussion on some statistical terms: F-ratio, error variance

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What is an F-ratio? Define all the technical terms in your answer.
What is error variance and how is it calculated?
Why would anyone ever want more than two (2) levels of an independent variable?
If you were doing a study to see if a treatment causes a significant effect, what would it mean if within groups variance was higher than between groups variance? If between groups variance was higher than within groups variance? Explain your answer
What is the purpose of a post-hoc test with analysis of variance?
What is probabilistic equivalence? Why is it important?

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What is an F-ratio? Define all the technical terms in your answer.
Answer: F ratio is the division of two variances from two independent samples. Usually, the formula is F=s12/s22 where s12 and s22 are two sample variances from two samples. And F-ratio is used to test the independence of two samples.

What is error variance and how is it calculated?
Answer: error variance is a measurement to show how far the actual data is far from the fitted data. Usually, the formula is error variance=total variance-systematic variance.

Why would anyone ever want more than two (2) levels of an independent ...

Solution Summary

The solution discusses some statistical terms: F-ratio, error variance, independent variable, between groups variance, within groups variance, post-hoc test and probabilistic equivalence.

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The goal is to test the functional dependence (prediction) of FVC on elderly subjects based on their height.

The goal is to test the functional dependence (prediction) of FVC on elderly subjects based on their height.

To test the regression assumption that there is a linear relationship between height and FVC and that the residuals in FVC are normally distributed and have equal variance along the linear relationship with height, we examined a histogram, a P-P plot of Regression Standardized residual, and a scatterplot. All graphs showed a normally distributed linear relationship.

Further, a linear regression analysis was performed to test whether the proportion of variance in FVC is statistically significant. It was determined that the proportion of variance in the FVC was significant (F= 699.4, df = 1, p<.001) and that the correlation of the height variable and FVC had a predictive ability.

Table 3. ANOVA for the Regression Equation, Height (cm) on Forced Vital Capacity (L)

Table 3 shows the computed proportion of variance in FVC. The residual sum of squares of the FVC (328.90) subtracted from its total variability (617.18) gives the amount of FVC predicted variance (288.28). The variance of FVC that is predicted (288.28) divided by the total FVC variability (617.18) gives the coefficient of determination (.467). The F ratio (699.43) provides the test of statistical significance.

Sum of Squares df Mean Square F
Regression 288.28 1 288.28 699.43**
Residual 328.90 798 .41
Total 617.18 799
** p < 0.01

- the regression equation and 95% CI for the Height coefficient with an accompanying narrative

The 95% CI of the actual predicted FVC value is between .057 to .066 and was calculated as:
95% CI for Height coefficient = unstandardized coefficient (Height) &#61617; (1.96 x Std. Error)
= .062 + or - (1.96 x .002)
Y= FVC
Constant = -7.193
B1 = .683
E= .385

o an example prediction using the regression equation for height = 160 cm, to include a confidence interval around the individual predicted value using the standard error of the estimate (see Model Summary)

Model Summary(b)

Model R R Square Adjusted R Square Std. Error of the Estimate
1 .683(a) .467 .466 .64199
a Predictors: (Constant), Height (cm)
b Dependent Variable: Forced Vital Capacity (L)

? Discussion of the outliers and their impact on the model, to include
o a comparison table (Outliers Included vs. Outliers Excluded) that includes all Model Summary statistics (R, R2, Adjusted R2, Std Error of the Estimate) and the two different regression equations

Model Summary

Model R Adjusted R Square Std. Error of the Estimate
outlier ~= 1.00 (Selected)
1 .723(a) .523 .522 .59941
a Predictors: (Constant), Height (cm)

Y = b0 + b1X + e
FVC = unstandardized coefficient (Constant) + unstandardized coefficient (Height) x Height

Descriptive Statistics
Mean Std. Deviation
Forced Vital Capacity (L) 2.9618 .87888
Height (cm) 164.8108 9.74889

See attached file for full problem description.

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