91Ó°ÊÓ

Hydrogen content is conjectured to be an important factor in porosity of aluminum alloy castings. The article "The Reduced Pressure Test as a Measuring Tool in the Evaluation of Porosity/Hydrogen Content in A1-7 Wt Pct Si-10 Vol Pct SiC(p) Metal Matrix Composite" (Metallurg. Trans., 1993: 1857-1868) gives the accompanying data on \(x=\) content and \(y=\) gas porosity for one particular measurement technique. $$ \begin{array}{l|lllllll} x & .18 & .20 & .21 & .21 & .21 & .22 & .23 \\ \hline y & .46 & .70 & .41 & .45 & .55 & .44 & .24 \\ x & .23 & .24 & .24 & .25 & .28 & .30 & .37 \\ \hline y & .47 & .22 & .80 & .88 & .70 & .72 & .75 \end{array} $$ MINITAB gives the following output in response to a CORRELATION command: Correlation of Hydrcon and Porosity \(=0.449\) a. Test at level \(.05\) to see whether the population correlation coefficient differs from 0 . b. If a simple linear regression analysis had been carried out, what percentage of observed variation in porosity could be attributed to the model relationship?

Step by step solution

01

Set Up Hypotheses

We want to test if the population correlation coefficient differs from zero. The null hypothesis, \(H_0\), is that the correlation coefficient \(\rho = 0\). The alternative hypothesis, \(H_1\), is that \(\rho eq 0\).

02

Determine Test Statistic

The sample correlation coefficient is given as \(r = 0.449\). Using a t-test for correlation, the test statistic \(t\) is calculated as follows: \[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \] where \(n\) is the number of data pairs, \(n = 14\). Thus, \[ t = \frac{0.449\sqrt{14-2}}{\sqrt{1-0.449^2}} \].

03

Calculate Test Statistic

Substitute the values into the formula: \[ t = \frac{0.449\sqrt{12}}{\sqrt{1-0.201601}} \] \[ t \approx \frac{0.449 \times 3.464}{0.8944} \approx \frac{1.554}{0.8944} \approx 1.737 \].

04

Determine Critical Value and Conclusion

For a two-tailed test at significance level \(\alpha = 0.05\), with \(n-2 = 12\) degrees of freedom, the critical t-value is approximately 2.179 (from a t-distribution table). Since \(1.737 < 2.179\), we fail to reject the null hypothesis \(H_0\). There is not enough evidence to conclude that the population correlation coefficient differs from zero.

05

Determine Coefficient of Determination (\(R^2\))

To find the percentage of observed variation explained by the model, use the formula \(R^2 = r^2\). Calculate \(R^2 = 0.449^2 \approx 0.201601\).

06

Interpret \(R^2\)

The coefficient of determination \(R^2\), which is approximately 0.202, indicates that 20.2% of the observed variation in porosity can be attributed to the model relationship.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Testing

Statistical testing is a way of making decisions or inferences about a population, based on a sample. When it comes to correlation, statistical testing helps us determine if there is a significant connection between two variables. In our exercise, we set up a hypothesis test to check whether the correlation coefficient between hydrogen content and gas porosity in aluminum alloy castings differs from zero.

- **Null Hypothesis ( $H_0$):** Indicates that there is no correlation, or $ ho = 0$. - **Alternative Hypothesis ( $H_1$):** Indicates that there is a correlation, or $ ho eq 0$. To evaluate the correlation, we used a t-test for the given sample correlation coefficient of 0.449 and verified it against the critical t-value for 12 degrees of freedom at a 0.05 significance level. Because our calculated t-value was less than the critical value, we concluded there wasn't enough evidence to suggest a significant correlation exists between the variables. This means we failed to reject the null hypothesis.

Linear Regression

Linear regression is a statistical tool used for understanding the relationship between two continuous variables. In this scenario, the relationship was examined between hydrogen content and gas porosity in the casting process. The simplest form of linear regression is the simple linear regression, which fits a straight line through the data.

The model can be represented by the equation:\[ y = mx + c \]where:- \(y\) is the dependent variable (gas porosity),- \(x\) is the independent variable (hydrogen content),- \(m\) is the slope of the line,- \(c\) is the y-intercept.By fitting this model, we aim to predict or explain porosity changes based on changes in hydrogen. Our regression analysis helps us identify trends and make forecasts based on our data. It's crucial to note, however, even with a given correlation, external variables might also affect outcomes. Hence, linear regression models demand careful interpretation.

Coefficient of Determination

The coefficient of determination, denoted as \(R^2\), tells us how well our statistical model explains the variation in the observed data. It's essentially the squared value of the correlation coefficient and is expressed as a percentage.
For our exercise, \(R^2\) was calculated as 0.202, meaning that approximately 20.2% of the variability in gas porosity is accounted for by the hydrogen content model.

This value provides insights into the strength of the relationship. A higher \(R^2\) means more variation is explained by the model, signaling a stronger connection. However, an \(R^2\) of 20.2% implies that while some correlation exists, a substantial amount of the variation is influenced by factors not included in the model. So, while helpful, \(R^2\) should be considered in conjunction with other analytical methods and domain knowledge for decision-making.