From this, we conclude that income does have a useful effect because this average difference will be multiplied by thousands of potential donors, leading to thousands of dollars overall. The 95% confidence interval for this average donation increase extends from $1.33 to $2.73, per $10,000 of neighborhood income. But how important is the effect of income on donation, practically speaking, in terms of dollar amounts? The regression coefficient, b = 0.000203, says that for each $10,000 increase in neighborhood income, we expect to see an increase of 0.000203 × 10,000 = $2.03 in the average donation size. In this case, neighborhood income explains only 3.2% of the variation in donations (using the coefficient of determination, R 2 = 0.032), which is small but to be expected given that donors' incomes vary even within a neighborhood (where they cannot be explained by average neighborhood income) and also given that donor behavior may differ even when their incomes are identical. How useful is this very highly significant result? With the large sample sizes available in data mining applications, it is often possible to detect statistical significance for a small effect (ie, a small effect that can be distinguished from randomness but may or may not be useful). There seems to be a positive relationship with considerable randomness, with people from higher-income neighborhoods donating larger amounts on average. Scatterplot with least-squares line to predict donation amount from neighborhood per capita income for donors. The standard error of the regression coefficient is S b = 0.20620, so the t statistic isįig. The t statistic approach gives the same answer, of course. You might also say that the slope coefficient is not significantly different from 0. There is no significant association between last week's and this week's market performance. Thus, we conclude that since 0 is contained in the interval, the apparent slope is not significant. This is a wide interval in fact, it contains 0, which would indicate no relationship.
We are 95% sure that the population slope β is between − 0.402 and 0.449. However, how accurately has this coefficient been estimated? The answer is provided by the confidence interval based on the estimate ( b = 0.02370), its standard error ( S b = 0.20620), and the critical t value 2.063899 with 26 – 2 = 24 degrees of freedom. The key is the slope coefficient, b = 0.02370, which says that only about 3% of last week's rise (or fall) will continue this week, on average. Should you believe this prediction equation? It pretends to help you forecast today's market behavior based on yesterday's (assuming that market behavior continues to act as though it is drawn from the same population).
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