Question

Music streaming services are the most popular way to listen to music. Data gathered over the last 12 months show Apple Music was used by an average of 1.86 million households with a sample standard deviation of 0.52 million family units. Over the same 12 months Spotify was used by an average of 2.34 million families with a sample standard deviation of 0.34 million. Assume the population standard deviations are not the same. Using a significance level of 0.05, test the hypothesis of no difference in the mean number of households picking either service.

Find the degrees of freedom for unequal variance test. (Round down your answer to nearest whole number.)

State the decision rule for 0.05 significance level: H0: μApple = μSpotify; H1: μApple ≠ μSpotify. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

Compute the value of the test statistic. (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)

Test the hypothesis of no difference in the mean number of households picking either variety of service to download songs.

Answer 1

df = (s1^2/n1 + s2^/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^/n2)^2/(n2 - 1))

    = ((0.52)^2/12 + (0.34)^2/12)^2/(((0.52)^2/12)^2/11 + ((0.34)^2/12)^2/11)

    = 19

At alpha = 0.05, the critical values are t^* = +/- 2.093

The test statistic t = ()/sqrt(s1^2/n1 + s2^/n2)

                             = (1.86 - 2.34)/sqrt((0.52)^2/12 + (0.34)^2/12)

                             = -2.676

Since the test statistic value is less than the negative critical value(-2.676 < -2.093), so we should reject the null hypothesis.

So at 5% significance level we can conclude that there is a significant difference in the mean number of house holds picking either variety of service to download songs.