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(a) Given claim: Differs

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain an equality.

\(\begin{array}{l}{H_0}:{\mu _1} = {\mu _2}\\{H_a}:{\mu _1} \ne {\mu _2}\end{array}\)

Determine the pooled standard deviation:

\({s_p} = \sqrt {\frac{{\left( {{n_1} - 1} \right)s_1^2 + \left( {{n_2} - 1} \right)s_2^2}}{{{n_1} + {n_2} - 2}}} = \sqrt {\frac{{(8 - 1)4.{9^2} + (9 - 1)4.{6^2}}}{{8 + 9 - 2}}} \approx 4.7424\)

Determine the test statistic:

\(t = \frac{{{{\bar x}_1} - {{\bar x}_2}}}{{{s_p}\sqrt {\frac{1}{{{n_1}}} + \frac{1}{{{n_2}}}} }} = \frac{{18.0 - 11.0}}{{4.7424\sqrt {\frac{1}{8} + \frac{1}{9}} }} \approx 3.038\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of the Student's t distribution table in the appendix containing the t value in the row\(df = {n_1} + {n_2} - 2 = 8 + 9 - 2 = 15\):

\(0.002 = 2 \times 0.001 < P < 2 \times 0.005 = 0.010\)

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:

\(P < 0.05 \Rightarrow Reject {H_0}\)

There is sufficient evidence to reject the claim that the two concentration distribution means are equal.

(b)

Given claim: Differs

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain an equality.

\(\begin{array}{l}{H_0}:{\mu _1} = {\mu _2}\\{H_a}:{\mu _1} \ne {\mu _2}\end{array}\)

Determine the test statistic:

\(t = \frac{{\left( {{{\bar x}_1} - {{\bar x}_2}} \right) - \left( {{\mu _1} - {\mu _2}} \right)}}{{\sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} }} = \frac{{18.0 - 11.0 - 0}}{{\sqrt {\frac{{{{4.9}^2}}}{8} + \frac{{{{4.6}^2}}}{9}} }} \approx 3.026\)

Determine the degrees of freedom (rounded down to the nearest integer):

\(\Delta = \frac{{{{\left( {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} \right)}^2}}}{{\frac{{{{\left( {s_1^2/{n_1}} \right)}^2}}}{{{n_1} - 1}} + \frac{{{{\left( {s_2^2/{n_2}} \right)}^2}}}{{{n_2} - 1}}}} = \frac{{{{\left( {\frac{{4.{9^2}}}{8} + \frac{{4.{6^2}}}{9}} \right)}^2}}}{{\frac{{{{\left( {4.{9^2}/8} \right)}^2}}}{{8 - 1}} + \frac{{{{\left( {4.{6^2}/9} \right)}^2}}}{{9 - 1}}}} \approx 14\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Student's T distribution in the appendix containing the t-value in the row df=14 :

\(0.002 = 2 \times 0.001 < P < 2 \times 0.005 = 0.010\)

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:

\(P < 0.05 \Rightarrow Reject {H_0}\)

There is sufficient evidence to reject the claim that the two concentration distribution means are equal.