Question

The article "Urban Battery Litter" cited in Example 8.14 gave the following summary data on zinc mass (g) for two different brands of size D batteries:

Brand Sample Size Sample Mean Sample SD

Duracell 15 138.52 7.76

Energizer 20 149.07 1.52

Assuming that both zinc mass distributions are at least approximately normal, carry out a test at significance level .05 to decide whether true average zinc mass is different for the two types of batteries.

Short answer

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

Step-by-step solution

To finding the zinc mass is different for the two types of batteries.

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.

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

To finding the zinc mass is different for the two types of batteries.

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.