Question

As the following table shows, projections indicate that the percent of U.S. adults with diabetes could dramatically increase.

(a) Find the logarithmic model that best fits the data in the table, with t as the number of years after 2000. (Round each coefficient to three places after the decimal.) D(t) = (b) Use the model to predict the percent of U.S. adults with diabetes in 2042. US adults with Year Diabetes (percentage) (

2010	14.2
2015	19.2
2020	21.1
2025	24.2
2030	27.7
2035	30.1
2040	31.2
2045	32.1
2050	33.4

Answer 1

Let t denote the number of years after 2000, is t = Year - 2000. The response variable y denotes the percentage of US adults with Diabetes.

(a)

To the above data we fit the following logarithmic regression model.

y = b log(t) + e where b is the slope parameter and e is the random error associated with y.

Using R, we get the least squares estimate of b as b = 7.98. Thus the model is y = 7.98 log(t)

The following results are obtained.

lm(formula = y ~ log(t) - 1)

Residuals:
Min 1Q Median 3Q Max
-4.1818 -2.4187 0.5478 1.7172 2.1699

Coefficients:
Estimate Std. Error t value Pr(>|t|)
log(t) 7.9831 0.2441 32.71 8.32e-10 ***
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.436 on 8 degrees of freedom
Multiple R-squared: 0.9926,   Adjusted R-squared: 0.9917
F-statistic: 1070 on 1 and 8 DF, p-value: 8.322e-10

We can observe that the coefficient of determination R2 = 0.9926. This implies that about 99.26% of the total variation present in the data set can be explained by the logarithmic regression of the percentage of diabetic adults on log(year). Hence the fitting of this logarithmic equation is almost perfect.

(b)

in 2042, t = 42. Then the predicted value of y is y = 7.98 log(42) = 29.83.

In 2042, 29.83% of US adults are likely to be diabetic.