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Trace metals in drinking water affect the flavor and an unusually high concentration can pose a health hazard. Twelve locations were selected and pairs of data were taken measuring zinc concentration in bottom water and surface water. (\\(\\mu g/l\\))

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The data is presented below:

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
123456789101112
Surface Water{r1[0]}{r1[1]}{r1[2]}{r1[3]}{r1[4]}{r1[5]}{r1[6]}{r1[7]}{r1[8]}{r1[9]}{r1[10]}{r1[11]}
Bottom Water{r2[0]}{r2[1]}{r2[2]}{r2[3]}{r2[4]}{r2[5]}{r2[6]}{r2[7]}{r2[8]}{r2[9]}{r2[10]}{r2[11]}
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It is believed that the bottom water will contain more zinc per litre than the surface water.

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In this example we are dealing with paired data.

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\\(H_0:\\) \\(\\mu_d=0\\)    i.e. The mean zinc concentration for surface water and bottom water is the same.

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\\(H_1:\\) \\(\\mu_d<0\\)    i.e  The mean zinc concentration for surface water is less than the mean zinc concentration for bottom water.

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We must evaluate the differences: for each pair of values \\(d=x_1-x_2\\)

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\\(\\var{diff}\\)

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We now have a sample of \\(n=\\var{sample_size}\\) differences.

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Recall:

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the formula for the mean difference:    \\(\\overline{d}=\\frac{\\sum {d}}{n}=\\var{diff_mean}\\)

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the formula for the standard deviation:   \\(s=\\sqrt{\\frac{\\sum{(d-\\overline{d})^2}}{n-1}}=\\var{diff_stdev}\\)

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the formula for the t-statistic:   \\(t=\\frac{\\overline{d}}{\\frac{s}{\\sqrt{n}}}=\\frac{\\var{diff_mean}}{\\frac{\\var{diff_stdev}}{\\sqrt{12}}}=\\var{test_statistic}\\)

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The t-table value will be for a one-tailed test and will have \\(n-1=11\\) degrees of freedom. Because of the alternative hypothesis the t-value chosen will be negative. 

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\\[\\begin{array}{r|rrrr}&0.10&0.05&0.01\\\\\\hline11&-\\var{t90}&-\\var{t95}&-\\var{t99}\\end{array}\\]

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Compare the test statistic with the t-table values and choose your conclusion.

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Input the appropriate sample mean: [[0]]

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Input the appropriate sample standard deviation: [[1]]

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Enter the value for the test statistic: t = [[2]]

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Having compared your test statistic with the table value, select one of the following conclusions that best describes your conclusion.

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Reject the Null Hypothesis and conclude that the mean zinc concentration is greater for bottom water than surface water

", "

Reject the Null Hypothesis at the 5% significance level but accept the Null Hypothesis at the 1% significance level and conclude that the mean zinc concentration is the same for bottom water and surface water

", "

Reject the Null Hypothesis at the 10% significance level but accept the Null Hypothesis at the 5% significance level and conclude that the mean zinc concentration is the same for bottom water and surface water

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Accept the Null Hypothesis at the 10% significance level and conclude that the mean zinc concentration is the same for bottom water and surface water

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