Chapter 6 Project - Music Kayne West wants to know the true proportion of the Ra
ID: 2932369 • Letter: C
Question
Chapter 6 Project - Music Kayne West wants to know the true proportion of the Raleigh population that listens to rap music. A recent survey asked the question, "What genre of music do you listen to most often?" Using only the data provided from the survey (see Excel file "Chapter 6 Project Data")answer the following questions with a 95% level of confidence. 1. Which one value is your best estimate of the true proportion of the Raleigh population that listens to rap music? (10 pts) 2. Is it likely that your answer to question 1 is the exact value of the true proportion? Explain using complete sentences. (10 pts) 3. Between which values is the proportion of the Raleigh population that listens to rap music likely to fall? Show all work including any Excel functions used. (20 pts) 4. How confident are you in this answer? Explain what this means to someone who has never heard of confidence in statistics (perhaps, Kanye). Hint: Do not interpret the confidence interval (20 pts) 5. How would your answer to question 3 change if you were given twice as many data points? Explain using complete sentences. (10 pts) 6. Kanye believes that the true proportion of the Raleigh population that listens to rap music is 40%. a. Does this agree or disagree with your answer? (10 pts) b. Is it likely that Kanye is correct? Explain using complete sentences. (10 pts) C. Is it possible that Kanye is correct? Explain using complete sentences. (10 pts)Explanation / Answer
From the above dataset shown in the above excel screenshot, 13 are 'Rap' out of 70 preferences.
1) Therefore the sample proportion of "Rap" music preference of the Raleigh population is p=13/70. Since sample proportion is an unbiased estimate of population proportion, therefore an estimate for the population proportion is 13/70.
2) No, it is not the exact value of the population proportion. It is an estimate with a standard error = standard deviation of sample proportion = sqrt[ p*(1-p)/n ] which specify the possible variations in the estimate.
3) Here the sample mean, p=13/70 = 0.1857143 and the standard error of estimate, se(p)=sqrt(p*(1-p)/70) = sqrt[ (13*57)/(70*70*70) ] = 0.0464796. Therefore the 95% confidence interval for the population proportion is given by,
[ p-1.96*se(p), p+1.96*se(p) ] = [ 0.1857143-1.96*0.0464796, 0.1857143+1.96*0.0464796 ] = [ 0.0946143, 0.276814].
4) The answer is given with a certainty that the population proportion lying inside this interval will happens with probability 0.95. This means that out of 100 different samples taken from that population, 95 times the sample mean will lie inside this interval.
5) If the sample proportion remains same in the double size sample, then estimate of population proportion remains same, but the standard error becomes different which will be se1(p)=sqrt(p*(1-p)/(2*70)) = se(p)/sqrt(2). So the changed answer will be [ 0.1857143-1.96*0.0464796/1.4142, 0.1857143+1.96*0.0464796/1.4142 ] = [ 0.121297, 0.250132 ].
6) Kanye believes that 40% of the poulation prefer Rap. But the sample proportion is hat(p)=13/70=18.57%.
a) The result shows the disagreement of the sample proportion with the hypothetical value. Although it is not possible to justify this agreement depending only on these two values.
b) The value 0.4 is not likely since from the confidence interval shown in question(3), it is evident that 0.4 lies outside the range shown.
c) To check that, we need to test the hypothesis, H0:p=0.4 vs the alternative H1: p not equal to 0.4.
To test that, from a large sample, the test statistic is defined as, z=(p-0.4)/se(p) = -4.61. Since |z|>z0.975 or z0.995 where za is upper 'a' point of standard normal distribution, the null hypothesis is rejected and it is concluded that Kanye is not correct.