Question

An article describes a method of analyzing digital music files to determine the key in which the music is written. In a sample of 307 pop music selections, the key was identified correctly in 242 of them. In a sample of 347 new-age selections, the key was identified correctly in 304 of them. Can you conclude that the method is more accurate for new-age songs than for pop songs? Find the P-value and state a conclusion. The P-value is Round the answer to four decimal places. We (Click to select) v conclude that the method is more accurate for new age songs than for pop songs. (Click to select) can cannot

Answer 1

For sample 1, we have that the sample size is N1 = 307, the number of favorable cases is X1 = 242, so then the sample proportion is Ôi = * = 302 = 0.7883 For sample 2, we have that the sample size is N2 = 347, the number of favorable cases is X2 = 304, so then the sample proportion is P2 X2 304 = 0.8761 347 - NO The value of the pooled proportion is computed as p = X1+X2 Ni+N2 242 304 307+347 = 0.8349 Also, the given significance level is a = 0.05. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = P2 Ha: P1 < P2 This corresponds to a left-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a left-tailed test is ze = -1.64. The rejection region for this left-tailed test is R = {z:z< -1.64} (3) Test Statistics The Z-statistic is computed as follows: N P1 - 2 VÕ(1 - 7)(1/nı +1/n2) 0.7883 – 0.8761 0.8349 . (1 – 0.8349)(1/307 +1/347) - 3.018 (4) Decision about the null hypothesis -3.018 < ze = -1.64, it is then concluded that the Since it is observed that z null hypothesis is rejected. Using the P-value approach: The p-value is p= 0.0013, and since p= 0.0013< 0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1 is less than P2, at the 0.05 significance level.

The p-value is 0.0013

We can conclude that the method is more accurate for new age songs than for pop songs.