Problem 4. An experiment was carried out to study the variation of the specific heat H
in calories per gram of a certain compound with T, its temperature in degrees Celsius. The
Summary statistics for the data are:
Xt = 900,

Xh = 20.16,
Xt
2 = 71000,
Xh
2 = 33.8894,

Xth = 1519.9.

(i) Obtain the analysis of variance (ANOVA) table.
(ii) Hence, test the hypothesis that the slope parameter is equal to zero, and comment on your
result.

### 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.

---

### 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.

---

### 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.

---

### 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?

Problem 4. An experiment was carried out to study the variation of the specific heat H in calories per gram of a certain compound with T, its temperature in degrees Celsius. The summary statistics for the data are: Xt = 900, Xh = 20.16, Xt2 = 71000, Xh2 = 33.8894, Xth = 1519.9. (i) Obtain the analysis of variance (ANOVA) table. (ii) Hence, test the hypothesis that the slope parameter is equal to zero, and comment on your result.

Explore how to calculate the ANOVA table and test the hypothesis that the slope parameter is zero in a linear regression model for specific heat and temperature data. Includes step-by-step derivation of SST, SSR, SSE, and hypothesis testing at a significance level.

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