On the applet, there is an option to see the Regression Table. The first row presents information from the two-sided test of inference for the y-intercept coefficient, and the second row presents similar information for the two-sided test of inference for the slope coefficient in the model. According to the table, the t-test statistic for the inference test for the slope turned out to be:

Question

BlackTom AI · Accepted Answer

The question asks about the t-test statistic for the inference test of the slope coefficient in a regression model, as reported in the Regression Table on the applet.

First, recall how the t-statistic for a slope is formed: t = (estimated slope coefficient) / (standard error of the slope). This statistic follows a t-distribution with degrees of freedom equal to n − 2, where n is the number of observations used to fit the simple linear regression model.

Next, interpret the sign of the t-statistic. The sign of t matches the sign of the estimated slope coefficient. If the slope is negative, the t-statistic will be negative as long as the standard error is positive (which it must be). Therefore, a negative t value indicates that the estimated slope is negative.

In reading the Regression Table, you would locate the row corresponding to the slope coefficient (often labeled as Beta1, x1, or simply the slope) and note two key numbers: the estimated slope (b1) and its standard error (SE_b1). The t-statistic is the quotient b1 / SE_b1. The table may also provide the p-value associated with this t-statistic, which you would compare to your chosen significance level to assess statistical significance.

Since the prompt mentions the first row contains the intercept and the second row contains the slope, you should isolate the second row to obtain the slope’s t-statistic. If the slope appears with a negative coefficient and a positive standard error, you should expect a negative t-statistic.

If the applet’s table shows an exact t-statistic value, you could cross-check by computing t from the displayed slope and its standard error: t = slope / SE. Any discrepancy would suggest either rounding differences or a mismatch between the displayed numbers and the underlying model.

Note on the absence of answer options: there are no multiple-choice options provided here. In practice, you would compare the computed t-statistic to the critical t-value for your degrees of freedom (e.g., t_crit ≈ ±t_{alpha/2, n-2}) or use the p-value to determine significance. Any given numeric value like -7.42 would be interpreted in this framework as indicating a fairly strong negative slope effect if the degrees of freedom and p-value align appropriately.

In summary, the t-statistic for the slope is derived from the ratio of the slope estimate to its standard error, its sign reflecting the slope direction, and its magnitude indicating how many standard errors the estimate lies from zero. To confirm the exact value, read the slope row in the regression table and perform the division if needed; otherwise rely on the applet’s reported t-statistic for that row.

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