Question

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1−1 and n2−1.)

Are Quarters Now Lighter? Weights of quarters are carefully considered in the design of the vending machines that we have all come to know and love. Data Set 29 “Coin Weights” in Appendix B includes weights of a sample of pre-1964 quarters (n = 40, x = 6.19267 g, s = 0.08700 g) and weights of a sample of post-1964 quarters (n = 40, x = 5.63930 g, s = 0.06194 g).

a. Use a 0.05 significance level to test the claim that pre-1964 quarters have a mean weight that is greater than the mean weight of post-1964 quarters.

b. Construct a confidence interval appropriate for the hypothesis test in part (a).

c. Do post-1964 quarters appear to weigh less than before 1964? If so, why aren’t vending machines affected very much by the difference?

Short answer

a) There is sufficient evidence to support the claim that pre-1964 quarters have mean weight greater than post-1964 quarters.

b) The 90% confidence interval is\(0.52492\;{\rm{g}} < \left( {{\mu _1} - {\mu _2}} \right) < 0.58182\;{\rm{g}}\)

c) Yes, pre-1964 quarters are mostly out of circulation. Hence, vending machines are not affected very much by the difference.

Step-by-step solution

Step 1: Given information

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, \(\alpha = 0.05\).

Step 2: State the null and alternative hypotheses

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.

Compute the test statistic

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.

Compute the critical value

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.

Compute the margin of error

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.

Compute the confidence interval

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.

Analyze the result

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.