Question

More time on the Internet: A researcher polled a sample of 1080 adults in the year 2010, asking them how many hours per week they spent on the Internet. The sample mean was 9.62 with a standard deviation of 13.27. A second sample of 1037 adults was taken in the year 2012. For this sample, the mean was 10.70 with a standard deviation of 14.46. Assume these are simple random samples from populations of adults. Can you conclude that the mean number of hours per week spent on the Internet increased between 2010 and 2012? Let μ1 denote the mean number of hours spent on the Internet in 2010. Use the a=0.05 level and the P-value method with the TI-84 calculator.

Part 2:

find the P-value:

Answer 1

Hypotheses : To test null hypothesis Ho :Mi - M2 = 0 against alternative hypothesis H1 :M1-42 < 0 This is a one tailed test as the alternative hypothesis contains '<' sign. Here, sz 13.272 14.462 21 = 9.62, si = 13.27, ni 1080, T2 = 10.70, S2 = 14.46, n2 = 1037, Standard error of mean = + = 0.6039 ni n2 1080 1037 The test statistic can be written as: (21 – 22) - (0) t- + ni n2 which under Ho follows a t distribution with 2081 df. where S2 13.272 1080 + 14.462 1037 ni n2 df 2081 13.272 -) 14.462 -)2 1080 + + 1037 1037-1 1080-1 n1-1 n2-1 and standard error for the difference between means si sź SE + = 13.272 1080 + 14.462 1037 = 0.603888 ni n2

Decision rule / Rejection region : We reject Ho at 0.05 level of significance if P-value < 0.05 or if tobs <-+0.05,df = -10.05,2081 = -1.645586 0.4- 0.3 density 0.2- 0.1- 0.0- 1 -4 -2 1 0 1 2 4 t Now, Value of the test statistic: The value of the test statistic is (9.62 – 10.70) tstat =-1.788412 -1.788 13.272 1080 14.462 1037 + associated degrees of freedom = 2081 The critical value = tcritical = - t0.05,2081 =-1.645586 ~ -1.646 P value = P(t2081 < tobs) = P(t2081 < -1.788412) = 0.036927 – 0.0369 Conclusion : Since p-value < 0.05 and tobs <tcritical = -1.646, so we reject Ho at 0.05 level of significance Hence, we can conclude that mean for first population is significantly less than mean for second population.

There is sufficient evidence to conclude that the mean number of hours per week spent on the Internet increased between 2010 and 2012.