Question

A musician is writing a new song and wants to know at which tempo he should write it to break onto the pop music charts. He takes a random sample of 100 pop songs and finds an average tempo of 131.9 bpm with a standard deviation of 14. Where applicable, use 3 decimals in the following answers. Please carry your intermediate calculations out to 4 decimal places.

a) Find the upper limit for a 95% confidence interval for the average beats per minute of a pop song.

b) What does the 95% confidence mean?

95% of intervals produced using this method will contain the true average beats per minute.

95% of the results from the sample will be valid for estimating the true average beats per minute.

    95% of samples will produce an interval containing this sample average number of beats per minutes.

95% of the sampled tempos will fall within this interval.

c) A producer asks the musician to produce a new interval. In particular, he wants the estimate to be within 1.33 bpm of the true average with 95% confidence. Assuming the standard deviation remains unchanged, how many songs would the musician need to sample?

Answer 1

a) The upper limit for a 95% confidence interval for the average beats per minute of a pop song is ,

Upper limit

14 = X + Zal2 Toni = 131.9 +1.96 * 131.9 +2.744 = 134.6440 100

Where , ; From Z-table

Za a/2 20.05/2 1.96

b) The 95% of intervals produced using this method will contain the true average beats per minute.

c) Since , margin of error=E=1.33

The required sample size is ,

$n=(\frac{Z_{\alpha/2}*\sigma}{E})^2=(\frac{1.96*14}{1.33})^2\approx 426$