Question

The table shows the wavelength of the sound produced by keys on a piano x keys away from the A above middle C.

A 2-column table with 5 rows. The first column is labeled number of keys above the a above middle c with entries 0, 2, 3, 6, 10. The second column is labeled wavelength with entries 78.41, 69.85, 65.93, 55.44, 44.01.
Using the exponential regression model, which is the best prediction of the wavelength of the key that is 8 above the A above middle C?

49.31 cm
49.44 cm
49.73 cm
49.78 cm

Answer 1

Use y = a·b^x. Take natural logs and do a linear regression on (x, ln y). That gives ln y ≈ 4.36210 − 0.05778x, so for x = 8:

ln ŷ = 4.36210 − 0.05778(8) = 3.89986 → ŷ = e^{3.89986} ≈ 49.44 cm.

Answer: 49.44 cm.