Consider A Regression Case In Which Y Is Hardness Of Mold ✓ Solved

Consider A Regression Situation In Which Y Hardness Of Molded

Consider a regression situation in which y = hardness of molded plastic and x = amount of time elapsed since termination of the molding process. Summary quantities included the following: n = 16, SSResid = 1236.43, and SSTo = 25,320.368.

(a) Calculate a point estimate of σ. (Give the answer to three decimal places.) On how many degrees of freedom is the estimate based?

(b) What proportion of observed variation in hardness can be explained by the simple linear regression model relationship between hardness and elapsed time? (Give the answer to three decimal places.)

Paper For Above Instructions

In this paper, we will analyze a regression situation where we aim to understand the relationship between the hardness of molded plastic (Y) and the elapsed time since the termination of the molding process (X). The data provided include a sample size (n) of 16, a residual sum of squares (SSResid) of 1236.43, and a total sum of squares (SSTo) of 25,320.368.

Point Estimate of σ

To calculate the point estimate of σ (the standard deviation of the residuals), we first need to determine the degrees of freedom associated with the residuals. The degrees of freedom for the residuals in a linear regression model are calculated using the formula:

Degrees of Freedom (df) = n - k

where n is the number of observations and k is the number of parameters estimated (including the intercept). In this case, since we have a simple linear regression (i.e., one predictor), k = 2 (the slope and the intercept). Thus, we can calculate the degrees of freedom:

df = 16 - 2 = 14

Now, to calculate the point estimate of σ, we utilize the residual mean square (MSE), which is derived from the SSResid:

MSE = SSResid / df

Now substituting the values:

MSE = 1236.43 / 14 ≈ 88.304

To find the point estimate of σ, we take the square root of the MSE:

σ ≈ √88.304 ≈ 9.396

Thus, the point estimate of σ is approximately 9.396 when rounded to three decimal places, it becomes 9.396.

Degrees of Freedom

The estimate of σ is based on 14 degrees of freedom, as calculated above.

Proportion of Observed Variation Explained by the Regression Model

To determine the proportion of observed variation in hardness that is explained by the regression model, we compute the coefficient of determination (R²). This can be found using the following formula:

R² = 1 - (SSResid / SSTo)

Substituting our values into the equation gives:

R² = 1 - (1236.43 / 25320.368) ≈ 1 - 0.0488 ≈ 0.9512

Thus, when rounded to three decimal places, the proportion of observed variation explained by the simple linear regression model is approximately 0.951.

Conclusion

In summary, we have calculated the point estimate of σ to be 9.396 based on 14 degrees of freedom. Additionally, about 95.1% of the observed variation in the hardness of molded plastic can be effectively explained by the regression model, demonstrating a significant relationship between the elapsed time since the molding process and the resulting hardness of the molded plastic.

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