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The statistics for the weights of samples of pre-1964 quarters and post-1964 quarters are summarized below:

Pre-1964 quarters:

\(\begin{array}{l}{n_1} = 40\\{{\bar x}_1} = 6.19267\;{\rm{g}}\\{s_1} = 0.08700\;{\rm{g}}\end{array}\)

Post-1964 quarters:

\(\begin{array}{l}{n_2} = 40\\{{\bar x}_2} = 5.63930\;{\rm{g}}\\{s_2} = 0.06194\;{\rm{g}}\end{array}\)

Level of significance is, \(\alpha = 0.05\).

a. Null hypothesis:the weights of pre-1964 quarters and post-1964 quarters have equal means.

Alternative hypothesis:the pre-1964 quarters have a mean weight that is greater than the mean weight of post-1964 quarters.

Mathematically,

\(\begin{array}{l}{H_0}:{\mu _1} = {\mu _2}\\{H_1}:{\mu _1} > {\mu _2}\end{array}\)

Where\({\mu _1},{\mu _2}\)are the population mean weights for the pre-1964 and post-1964 groups respectively?

The test is right tailed.

The test statistic is given by,

\(\begin{array}{c}t = \frac{{\left( {{{\bar x}_1} - {{\bar x}_2}} \right) - \left( {{\mu _1} - {\mu _2}} \right)}}{{\sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} }}\\ = \frac{{\left( {6.19267 - 5.6393} \right) - \left( 0 \right)}}{{\sqrt {\frac{{{{0.087}^2}}}{{40}} + \frac{{{{0.06194}^2}}}{{40}}} }}\\ = 32.771\end{array}\)

Calculated value of test statistic is 32.771.

Degrees of freedom are:

\(\begin{array}{c}\left( {{n_1} - 1} \right) = \left( {40 - 1} \right)\\ = 39\end{array}\)

\(\begin{array}{c}\left( {{n_2} - 1} \right) = \left( {40 - 1} \right)\\ = 39\end{array}\)

Degrees of freedom is a minimum of two values, which is 39.

Refer to the t-table for the right-tailed test with a 0.05 level of significance with 39 degrees of freedom.

The critical value is given by,

\(\begin{array}{c}P\left( {t > {t_\alpha }} \right) = \alpha \\P\left( {t > {t_{0.05}}} \right) = 0.05\end{array}\)

Thus, the critical value is 1.685.

As the test statistic value is greater than the critical value, the null hypothesis is rejected. Thus, there is sufficient evidence to support the claim that the mean weight for pre-1964 quarters is greater than the mean weight for post-1964 quarters.

b. The 0.05 significance level for one-tailed test implies 90% level of confidence.

The margin of error is given by,

\(\begin{array}{c}E = {t_{\frac{\alpha }{2}}} \times \sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} \\ = 1.685 \times \sqrt {\frac{{{{0.087}^2}}}{{40}} + \frac{{{{0.06194}^2}}}{{40}}} \\ = 0.0284\end{array}\)

The margin of error is 0.0284.

Here, 1.685 is obtained from the t-table with 39 degrees of freedom and 0.10 level of significance.

The 90% confidence interval for two samples is given by,

\(\begin{array}{c}{\rm{Confidence}}\;{\rm{interval}} = \left( {{{\bar x}_1} - {{\bar x}_2}} \right) - E < \left( {{\mu _1} - {\mu _2}} \right) < \left( {{{\bar x}_1} - {{\bar x}_2}} \right) + E\\ = \left( {6.19267 - 5.6393} \right) - 0.02845 < \left( {{\mu _1} - {\mu _2}} \right) < \left( {6.19267 - 5.6393} \right) + 0.02845\\ = 0.5249 < \left( {{\mu _1} - {\mu _2}} \right) < 0.5818\end{array}\)

Therefore, the 90% confidence interval is between\(0.5249 < \left( {{\mu _1} - {\mu _2}} \right) < 0.5818\).

As 0 does not belong to the interval, there is enough evidence that the mean weight of pre-1964 quarters is greater than mean weight of post-1964 quarters.

c) From both the results, there is enough evidence that the mean weights of post-1964 quarters appear to be less than pre-1964 quarters, as the estimates are positive.

As the 1964 coins are out of circulation, the vending machine would not be affected by the difference.