Given Information:

• $X_t$ (mean of temperature, $T$) = 900
• $X_h$ (mean of specific heat, $H$) = 20.16
• $X_t^2$ (sum of squares of temperature, $T^2$) = 71,000
• $X_h^2$ (sum of squares of specific heat, $H^2$) = 33.8894
• $X_{th}$ (sum of products of temperature and specific heat, $T \times H$) = 1519.9

We need to address two parts of the problem:

1. ANOVA Table for the regression analysis.
2. Test the hypothesis that the slope parameter $\beta_1$ is equal to zero.

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Step 1: Calculate the required sums of squares and components for the ANOVA table.

Regression Equation:

We are assuming the linear regression model: $H = \beta_0 + \beta_1 T + \epsilon$ Where $H$ is the specific heat and $T$ is the temperature.

The ANOVA table will help assess the goodness of fit of the regression model. It will include the following components:

• Total Sum of Squares (SST): $SST = \sum (H_i - \bar{H})^2$ This is the variation in the observed values of $H$.

• Regression Sum of Squares (SSR): $SSR = \sum (\hat{H_i} - \bar{H})^2$ This represents the portion of total variation explained by the regression model.

• Residual Sum of Squares (SSE): $SSE = SST - SSR$ This represents the portion of variation unexplained by the regression model.

Let's start with the computation of the slope $\beta_1$ and intercept $\beta_0$.

Step 1: Slope ($\beta_1$) and Intercept ($\beta_0$)

Using the formula for the slope $\beta_1$ and the intercept $\beta_0$ for simple linear regression: $\beta_1 = \frac{n \sum T H - \sum T \sum H}{n \sum T^2 - (\sum T)^2}$ $\beta_0 = \bar{H} - \beta_1 \bar{T}$ Where $n$ is the number of data points, and $\bar{T}$, $\bar{H}$ are the sample means of $T$ and $H$, respectively.

Step 2: ANOVA Table Structure

The ANOVA Table will have the following columns:

1. Source of Variation: "Regression", "Residual", and "Total".
2. Sum of Squares (SS): The computed sums of squares (SST, SSR, SSE).
3. Degrees of Freedom (df):
   • Regression df = 1
   • Residual df = $n - 2$ (since one degree of freedom is lost for estimating the intercept and slope).
   • Total df = $n - 1$.
4. Mean Square (MS): $MS = \frac{SS}{df}$.
5. F-Statistic: $F = \frac{MS_{\text{Regression}}}{MS_{\text{Residual}}}$.

We need to compute each part to fill in the table.

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Step 2: Hypothesis Testing for the Slope

The hypothesis we are testing is:

• $H_0: \beta_1 = 0$ (No linear relationship between $T$ and $H$)
• $H_1: \beta_1 \neq 0$ (There is a linear relationship)

The test statistic for testing the slope is: $t = \frac{\hat{\beta_1}}{SE_{\hat{\beta_1}}}$ Where $SE_{\hat{\beta_1}}$ is the standard error of the slope estimate.

The standard error $SE_{\hat{\beta_1}}$ is computed as: $SE_{\hat{\beta_1}} = \sqrt{\frac{SSE}{(n - 2) \cdot \sum (T_i - \bar{T})^2}}$

We will test the hypothesis at a significance level $\alpha$ (commonly 0.05). If the p-value is less than $\alpha$, we reject $H_0$ and conclude that there is a significant linear relationship.

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To Proceed:

To calculate the ANOVA table and hypothesis test, we would need the number of data points, $n$, to proceed with the calculations. Could you provide the number of data points, or any additional details such as the raw data?