{"id":2570,"date":"2024-07-11T18:35:11","date_gmt":"2024-07-11T22:35:11","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&p=2570"},"modified":"2024-07-23T21:29:21","modified_gmt":"2024-07-24T01:29:21","slug":"hypothesis-testing-for-one-mean-examples","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/hypothesis-testing-for-one-mean-examples\/","title":{"raw":"Hypothesis Testing for One Mean","rendered":"Hypothesis Testing for One Mean"},"content":{"raw":"
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

In this section, we will perform left, right and two-tailed hypothesis tests for means from one sample.<\/div>\r\n<\/div>\r\nIn this section, we will step through several examples of how to perform hypothesis testing for sample means. Let us start by recapping the required formula presented in the previous section<\/a>:\r\n

Hypotheses<\/h2>\r\nH0<\/sub>: [latex]\\mu = \\mu_{original}[\/latex] (all tails)\r\n\r\nHA<\/sub>: [latex]\\mu < \\mu_{original}[\/latex] (left-tailed), [latex]\\mu > \\mu_{original}[\/latex] (right-tailed), [latex]\\mu \\neq \\mu_{original}[\/latex] (two-tailed)\r\n

Test Statistic<\/h2>\r\nThe test statistic formula is the same, regardless of what tailed test we are performing:\r\n\r\n\\[t_{test} = \\frac{\\bar{x}-\\mu}{\\frac{s}{\\sqrt{n}}}\\]\r\n

P-Values<\/h2>\r\nWe could use Excel's T.DIST() function to calculate all p-values but, it is slightly easier to use different functions depending on which hypothesis test we are performing (left, right or two-tailed):\r\n\r\n\\[p_{\\text{ left-tailed}}=\\text{T.DIST}(t_{test},n-1,\\text{TRUE})\\]\r\n\r\n\\[p_{\\text{ two-tailed}}=\\text{T.DIST.2T}(\\text{abs}(t_{test}), n-1) \\]\r\n\r\n\\[p_{\\text{ right-tailed}}=\\text{T.DIST.RT}(t_{test},n-1)\\]\r\n\r\nRemember:<\/strong> When use T.DIST.2T<\/a>, we must only input a positive t[latex]_{test}[\/latex] value. For this reason, we can use an absolute value call ABS()<\/a> to ensure that we input a positive value in the T.DIST.2T function.\r\n

Right-tailed Test (Exercise)<\/h1>\r\nLet us start by working through a right-tailed hypothesis test problem for a sample with one mean.\r\n

Example 57.1<\/h2>\r\nProblem Setup<\/span><\/strong>: Stats Canada has published a report stating that the average starting business student's starting salary is $5,400 per month. Your college's marketing team believes that their business diploma grads earn higher than $5,400, on average, upon graduation. A sample of 100 grads is polled and their starting salaries are reported. The average starting salary is reported at $6,500 with a standard deviation of $3,200.\r\n\r\n\"Image<\/a>\r\n\r\nQuestion<\/span><\/strong>: Is there sufficient evidence, based on the sample of 100 grads to support the marketing team's hypothesis at a 5% level of significance?\r\n\r\nYou Try<\/strong><\/span>: Scroll through the questions below to solve the hypothesis test problem above.\r\n
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\r\n\r\n[h5p id=\"181\"]\r\n\r\n<\/div>\r\n<\/div>\r\nSolutions<\/strong><\/span>: Click here<\/a> to download the Excel solutions and click on the sections below to reveal the written solutions to each part of the above exercise.\r\n
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Assumptions<\/h1>\r\n