# Chance (Fall 2004) published a study on modeling a certain pitch of a musical composition. Data on 147 famous compositions were used to model the numb

Modeling

Chance (Fall 2004) published a study on modeling a certain pitch of a musical composition. Data on 147 famous compositions were used to model the number of times (y) a certain pitch occurs called entropy- as a straight-line function of year of birth (x) of the composer. On the basis of the scatterplot of the data, the standard deviation $$\sigma \sigma$$ of the model is estimated to be $$s = .1$$. For a given year (x), about 95% of the actual entropy values (y) will fall within d units of their predicted values. Find the value of d.

2021-06-01

Given: $$s=1$$
The Empirical Rule or (68-95-99.7 rule) tells us: Approximately 68% of the observations are within 1 standard deviation of the mean.
Approximately 95% of the observations are within 2 standard deviations of the mean.
Approximately 99.7% of the observations are within 3 standard deviations of the mean.
By the empirical rule, about 95% of the data. values are within two standard deviations of the mean.
About 95% of the actual y-values are within 2s of the predicted y-values (due to the fact that the errors have approximately a, normal distribution with standard deviation s).
$$d=2 s=2(0.1)=0.2$$