Math Solutions

[All Steps] VCR Market Penetration The following data gives the market penetration of (percentage of households that own) VCRs in the U.S. Year % 1978 0.3

Question: VCR Market Penetration

The following data gives the market penetration of (percentage of households that own) VCRs in the U.S.

Year %
1978 0.3
1979 0.5
1980 1.1
1981 1.8
1982 3.1
1983 5.5
1984 10.6
1985 20.8
1986 36.0
1987 48.7
1988 58.0

A. Plot the data. There clearly must be a limiting value since there will never be more than a 100% of the homes that own VCRs.



B. Formulate a model that assumes maximum market penetration of 70%.

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 5.694082 0.116193 49.00529 3.08E-12 5.431234 5.956929
t -0.70585 0.01964 -35.939 4.95E-11 -0.75028 -0.66142

which implies that the model is

\[P\left( t \right)=\frac{70}{1+297.1038{{e}^{-0.70585t}}}\]


C. Test you model by plotting the predictions against the data.


D. In what year will maximum penetration be attained? What percentage of households would you expect in year 2000?

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