\nA research believes that the mean incom of workers in Akokwa town is 20,000 dollars. She wants to test this hypothesis against the alternative that the mean income is not equal to 20,000 naira. A random sample of 9 workers in the town is chosen, and their incomes (in dollars) turn out to be the following:<\/em>
\n24000, 13400, 18400, 22900, 13800, 8200, 11100, 9300, 14600.<\/em><\/p>\n
(a) If the sociologist set a 5 percent significance level, should she accept or reject this hypothesis?<\/em> <\/p>\n Solution<\/b><\/span> <\/p>\n <\/ins> H0<\/sub>: mean = 20,000 naira <\/p>\n Step 2: State the alpha<\/b> <\/p>\n Step 3: Calculate the degrees of freedom<\/b> <\/p>\n Step 4: State the decision rule (acceptance criteria)<\/b> <\/p>\n Decision rule<\/b>: Accept the null hypothesis if absolute value of t is less than 2.306 or greater than -2.306<\/p>\n We can do a little sketch to illustrate this in Figure 1.0. Note that the 5% has been divided into two parts of 2.5% on both sides of the graph<\/p>\n <\/p>\n Figure 1.0:<\/b> Sketch of Accept and Reject region<\/p>\n <\/p>\n 5. Calculate the test Statistic (Absolute value of t)<\/b>
\n(b) If she sets a 1 percent significance level, shoud she accept or reject the null hypothesis.<\/em><\/p>\n
\nWe would start with the first step and progress through the last step. These are the steps:<\/p>\n\n
\nStep 1: State the null and alternate hypothesis<\/b>
\nThe null hypothesis is the what is currently believed and needs to be tested. From the question the believe is that the mean income is 20,000 naira. Therefore<\/p>\n
\nH1<\/sub>: mean \u2260 20,000 naira<\/p>\n
\nFrom the question, we have 5% significance level. This is the same as alpha level of 0.05 ( or 95% confidence). So we can state the alpha level as:
\n\u03b1 = 0.05<\/p>\n
\nDegrees of freedom is given as df = n-1. So in this case:
\nDegrees of freedom df = 9 -1 = 8<\/p>\n
\nTo do this we need to look up the critical value of the t from table of t-distribution or t-Table.
\nIn the table, the alpha is placed in the column while the degrees of freedom are on the rows. In our case we look for df of 8 under 0.05. It is normally written as:
\nTcrit<\/sub> or T8, 0.05<\/sub> = 2.306<\/p>\n![](\"https:\/\/4.bp.blogspot.com\/-U-hlcrpNVXI\/WyrNa55hp6I\/AAAAAAAAB-s\/ddXQRk3hCwost5d5mcDubx9-RMKqL2H1gCLcBGAs\/s1600\/Accept-and-Reject%2BRegion.jpg\")
<\/a><\/div>\n
\nThe formula for the test statistic is given by:<\/p>\n