418 CHAPTER 8 Hypothesis Testing Here is an important note if using Table A-4 for finding critical values: In Table A-4, each critical value of x2 in the body of the table corresponds to an area given in the top row of the table, and each area in that top row is a cumulative area to the right of the critical value. CAUTION Table A-4 for the chi-square distribution uses cumulative areas from the right (unlike Table A-2 for the standard normal distribution, which provides cumulative areas from the left.) See Example 1 in Section 7-3. Minting Quarters EXAMPLE 1 U.S. Mint specifications require that quarters manufactured after 1964 have weights with a mean of 5.670 g and a standard deviation of 0.062 g. Listed below are weights (grams) of a simple random sample of quarters made with a new minting process designed to reduce the standard deviation so that weights of quarters are more consistent. Use a 0.05 significance level to test the claim that these weights are from a population with a standard deviation that is less than 0.062 g. 5.7424 5.7328 5.7268 5.5938 5.6342 5.6839 5.6651 5.6925 5.6803 5.6245 5.7985 5.7180 5.7299 5.6582 5.7360 5.6546 5.7222 5.6619 5.7041 5.6528 5.6210 5.6613 5.6484 5.6502 SOLUTION REQUIREMENT CHECK (1) The sample is a simple random sample. (2) In checking for normality, we see that the sample has no outliers, the accompanying normal quantile plot shows points that are reasonably close to a straight-line pattern, and there is no other pattern that is not a straight line. Both requirements are satisfied. Statdisk Technology Technology capable of conducting this test will typically display the P-value. Statdisk can be used as described at the end of this section, and the result will be as shown in the accompanying display. The display shows that the test statistic is x2 = 13.795 (rounded) and the P-value is 0.0674. Step 1: The claim that “the standard deviation is less than 0.062 g” is expressed in symbolic form as s 6 0.062 g. Step 2: If the original claim is false, then s Ú 0.062 g.
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