You manage a 4 year old web service. 4 months ago, you moved to a new layout for the website. It is desirable for the business that users spend more time on the website. You are to come up with a confidence interval for the change in the mean value of the time spent on the website. Let X1,X2,X3,,X1461 denote the average time spent by the users on the website every day in the 4 years (i. e. 1461 days) prior to the change in the layout. Let Y1,Y2,,Y122 denote the average time spent by the users on the website every day since the change of the layout. The data has the following statistics: X = 0.92, S = 0.17
X = 1.26, S = 0.11
(a) Finding confidence intervals
i. Using a z-table, come up with a two-sided 95%-confidence interval for muy - mux
ii. Using a z-table, come up with a two-sided 99%-confidence interval for muy - muX.
(b) Hypothesis testing:
You want to test the hypothesis mux = muy versus mux < muy with significance level alpha = 0.05.
i. Specify the critical region for the difference in the sample means, i. e. for what value of Y - X would you reject the hypothesis?
ii. Given the sample statistics from problem 2, i. e. X = 0.92 and Y = 1.26, what would your conclusion be? State your conclusion in technical terms (i. e. as if talking to a statistician).
iii. State your conclusion in everyday language (i. e. as if explaining to somebody interested in web design but not interested in statistics).
iv. What is the p-value of this statistic?
(c) We assumed the average daily use of customers over the 4 years was independent and identically distributed. Do you find this assumption reasonable? Why? If not, can you suggest a better way to perform this analysis?