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 127.5 bpm with a standard deviation of 12. 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 the sampled tempos will fall within this interval. 95% of the results from the sample will be valid for estimating the true average beats per minute. 95% of intervals produced using this method will contain the true average beats per minute, 95% of samples will produce an interval containing this sample average number of beats per minutes. c) A producer asks the musician to produce a new interval. In particular, he wants the estimate to be within 1.44 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) To find the upper limit for a 95% confidence interval for the average beats per minute of a pop song, we can use the formula:

Upper Limit = Sample Mean + (Z * (Standard Deviation / √n))

Where:

Sample Mean = 127.5 bpm

Z = Z-score for a 95% confidence level (which corresponds to a 2-tailed test) = 1.96 (obtained from the standard normal distribution table)

Standard Deviation = 12 bpm

n = Sample Size

Plugging in the values, we get:

Upper Limit = 127.5 + (1.96 * (12 / √100))

= 127.5 + (1.96 * (12 / 10))

= 127.5 + (1.96 * 1.2)

= 127.5 + 2.352

= 129.852

Therefore, the upper limit for a 95% confidence interval for the average beats per minute of a pop song is approximately 129.852 bpm.

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

c) To determine the sample size needed to estimate the true average beats per minute within 1.44 bpm with 95% confidence, we can use the formula:

n = ((Z * Standard Deviation) / Margin of Error)²

Where:

Z = Z-score for a 95% confidence level (which corresponds to a 2-tailed test) = 1.96 (obtained from the standard normal distribution table)

Standard Deviation = 12 bpm

Margin of Error = 1.44 bpm

Plugging in the values, we get:

n = ((1.96 * 12) / 1.44)²

= (23.52 / 1.44)²

= 16.333³

= 449.438

Therefore, the musician would need to sample at least 450 songs in order to estimate the true average beats per minute within 1.44 bpm with 95% confidence, assuming the standard deviation remains unchanged.

Step-by-step explanation: