91Ó°ÊÓ

Given in the output:

\(\begin{array}{l}{{\bar x}_1} = 138.52\\{s_1} = 7.76\\{{\bar x}_2} = 149.07\\{s_2} = 1.52{n_1} = 15\\{n_2} = 20\\\alpha = 0.05\end{array}\)

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:

\(t = \frac{{\left( {{{\bar x}_1} - {{\bar x}_2}} \right) - \left( {{\mu _1} - {\mu _2}} \right)}}{{\sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} }} = \frac{{138.52 - 149.07 - 0}}{{\sqrt {\frac{{7.7{6^2}}}{{15}} + \frac{{1.5{2^2}}}{{20}}} }} \approx - 5.191\)

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

\(\Delta = \frac{{{{\left( {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} \right)}^2}}}{{\frac{{{{\left( {s_1^2/{n_1}} \right)}^2}}}{{{n_1} - 1}} + \frac{{{{\left( {s_2^2/{n_2}} \right)}^2}}}{{{n_2} - 1}}}} = \frac{{{{\left( {\frac{{7.7{6^2}}}{{15}} + \frac{{1.5{2^2}}}{{20}}} \right)}^2}}}{{\frac{{{{\left( {7.7{6^2}/15} \right)}^2}}}{{15 - 1}} + \frac{{{{\left( {1.5{2^2}/20} \right)}^2}}}{{20 - 1}}}} \approx 14\)

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 :

\(P < 2 \times 0.0005 = 0.001\)

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

\(P < 0.01 \Rightarrow Reject {H_0}\)

There is sufficient evidence to support the claim that the true average zinc mass is different from the two types of batteries.