Question

Problem 9. A couple shares a digital music player that has a feature of randomly selecting songs to play. The couple is interested in determining statistically whether they have loaded the same proportion of songs into the player. Suppose that when the player is in the random-selection mode, 27 of the first 50 songs selected are songs loaded by the wife. Q1. (5 points) Construct the 90% confidence interval for the population proportion of songs loaded by the wife and write a brief interpretation of your confidence interval in terms of the context. Q2. Do the data provide sufficient evidence to support the claim that the couple have loaded different proportions of songs into the player at a significance level of 0.10? Answer the question by a) (3 points) formulating null and alternative hypotheses in terms of the population parameter as well as the context of this application. b) (7 points) finding the P-value of the test, interpreting your P-value in the context of the application, and making your conclusion

Answer 1

Answer! Q.9. tele are given that, 27 of the first so songs selected are Songs loaded by the wile. ine. here, n=50, x=27 He want to construed the go% confidence interval for the population proportion of songs loaded by the wife. He know that the contidence interval for population proportion i.e. P.is Å 72 P V n where, p is the sample proportion n is the sample size 2* is the appropriate value from the standard normal distribution for desired Confidence level. find here, Ê = 27 ist we SO to 0.54 ² = sample proportion of songs loaded by the wife 0.54 here z value for the go% Confidence level is 1.645 here n. So

Lower bound P- z* À CI- V 0.54 1.645 10.54 (1-0.54) 50 0-54 - 0.2484 SO -0.54 -(1.645 x 0.0705) - 0.54 - 0.1159725 Lower bound O. 4240 42.4% Upper bound P + 2* PCI- 0.54 + 10.54 (1-0.54] 50 = 0.54+ (1.645X0.0705) - -0.54 to. 1159725 0.6559725 Upper bound O. 6560 - 65. 60% go% Confidence interval for the population pooportion of songs doaded by the wife is Co. 424 < P < 0.656.) To interpret these results within the context of the problem, we can say that with 90% Confidence of songs loaded by the coile is somewhere between 142.4% and 65.6% given sample. based on

Date Page VS Q.2. We want to find Do the data provide sufficient evidence to Support the claim that the couple have loaded dik perent proportions of songs into the player significa icance level of o. 10? ay To test the claim. We set the hypothesis Ho! Couple have loaded same proportions of songs into the player. inl. P,= P2 Hi : Couple have loaded different proportions t of songs into the player inl. P, ZP2 Pia proportion of the wife's songs doaded P2 proportion of the husband's songs doaded, here songs doaded by wile out of so is 27 songs husband 0 = n2 = 50 toaded by • 50-29-23 To test the nell hypothesis Ho the test statistic is.. Pi Pi = 2 P2 Pleap) (1 + 1) where, Sample Proportion of is 751 sample P2 - Sample proportion of and sample sample size of ist Sample Sample size of and sample Overall proportion or pooled proportion. ni no 2 P -

Pada For sample 1. sample size number of favorable cases 21 27 sample Proportion is 27 50 0.54 for sample 2. 12-so- Sample size U2 - 23 number of favorable cases Sample Proportion is Rê - 0.46. X2 na 23 50 The value of the pooled proportion is Computed as, P = geitra 27+23 50 nitna in 50+50 100 Also the given significance level is a = 0,10 Test statistics is z = À - JP (1-P) (1 , +12) 0.54 0.46 1-0.5)

Dete Page 08 N 0.25 0.04 0.08 o.1 2 by. Po value: the p-value for the 2-score - 0.8 at 0.10 level of significance is 0.4237 & since P=0.42377, 0.10 it is concluded that hall hypothesis is not rejected. Conclusion: It is concluded that the hall hypothesis Ho is not rejected There hore there is not enough evidence to claim that the couple have loaded different proportions of songs into the Playen er at a significance level of 0.10 j.e. population proportion p, is different than P2 at the oilo significance level.