Question

12. In 2015, 88% of US residents used the internet, up from 14% in 1995 The table shows the percent who use the internet for selected years from 2000 and projected to 2025. Year 2000 2005 2010 2015 2020 2025 Percent 67 79 82 88 95 98 Find the logarithmic function that models the percent p as a function of x, the number of years after 1990. State the correlation coefficient. a) b) Use the model to predict the percentage of internet users in the United States in 2032. (Round your answer to one decimal place.)

Answer 1

Hi.

If x is the number of years after 1990 then 2000 is x=10 and so, we first make the following table:

x	p
10	67
15	79
20	82
25	88
30	95
35	98

And the logarithmic model will have the following structure:

$p=a+bLn(x)$

Then we need to make the following table to calculate a and b

x	p	Ln(x)=X1	X1mean	pmean	X1-X1mean	p-pmean	(X1-X1mean)^2	(p-pmean)^2	(X1-X1mean)(p-pmean)
10	67	2.302585	3.030298	84.83333	-0.727713046	-17.83333333	0.529566278	318.0277778	12.97754933
15	79	2.70805	3.030298	84.83333	-0.322247938	-5.833333333	0.103843734	34.02777778	1.879779639
20	82	2.995732	3.030298	84.83333	-0.034565866	-2.833333333	0.001194799	8.027777778	0.09793662
25	88	3.218876	3.030298	84.83333	0.188577686	3.166666667	0.035561544	10.02777778	0.597162671
30	95	3.401197	3.030298	84.83333	0.370899242	10.16666667	0.137566248	103.3611111	3.770808964
35	98	3.555348	3.030298	84.83333	0.525049922	13.16666667	0.275677421	173.3611111	6.913157309
						∑	1.083410023	646.8333333	26.23639453

Where:

Σ(ri-i)(pi-P) 26236 24225 108 = 24.225

And

a = p-b(F) = 84.833-(24.225) × 3.030= 11.431

a) So the model is:

p = 24.225Ln(x) + 11.431

The correlation coefficient can be found by excel:

y 24.2161n(x) +11.45 R2 0.9823 TiHi 80 40 20 20 10 15 25 漠) 35 40

R2-0,9823
       so
R0.991

b)

The year 2032 is x=42 so:

p = 24.225Ln(42) 11.431 = 102

Which basically means that in the year 2032 100% of U.S residents will have internet. The result of p can't be higher than 100