You are working for a company that manufactures cutting blades for several equipment manufacturers. For a specific type of blade, the desired blade length is 12.0 inches. As part of your quality control engineer job, your goal is to statistically determine the actual numbers out of a production lot. You take 25 random samples out of a production period (which produces hundreds of such blades) and make measurements on your sample to find your mean length is 11. 7 inches and the standard deviation of your samples is 0.6 inches.

a. What is the 95% confidence interval for the blade length for this production period?

b. What is the 90% confidence interval for the blade lengths for this production period?

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To complicate matters, statisticians list confidence intervals backwards by using a concept called a level of significance, or a where: a = 1-%c For instance, the level of significance for a 90% confidence level would be a=0.1. The columns of the t-tables are arranged for various critical points of this a value (e.g. 10%, 5%, 1% etc). The corresponding t-values for various degrees of freedom at these significance levels are then listed as elements of the table. Table 1 Single sided Student's t distribution table listing to2 versus v. (probability higher than the critical t-value. Example: the probability that we will have a value exceeding the critical t- value of 2.228 for a degree of freedom of 10 is 0.025) to.40 to.05 ta2 D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 to.25 23 24 25 26 to.10 to.025 to.01 to.005 5.4079 5.0413 0.324920 1.000000 3.077684 6.313752 12.70620 31.82052 63.65674 636.6192 0.288675 0.816497 1.885618 2.919986 4.30265 6.96456 9.92484 31.5991 0.276671 0.764892 1.637744 2.353363 3.18245 4.54070 5.84091 12.9240 0.270722 0.740697 1.533206 2.131847 2.77645 3.74695 4.60409 8.6103 0.267181 0.726687 1.475884 2.015048 2.57058 3.36493 4.03214 6.8688 0.264835 0.717558 1.439756 1.943180 2.44691 3.14267 3.70743 5.9588 0.263167 0.711142 1.414924 1.894579 2.36462 2.99795 3.49948 0.261921 0.706387 1.396815 1.859548 2.30600 2.89646 3.35539 0.260955 0.702722 1.383029 1.833113 2.26216 2.82144 3.24984 4.7809 0.260185 0.699812 1.372184 1.812461 2.22814 2.76377 3.16927 4.5869 0.259556 0.697445 1.363430 1.795885 2.20099 2.71808 3.10581 4.4370 0.259033 0.695483 1.356217 1.782288 2.17881 2.68100 3.05454 4.3178 0.258591 0.693829 1.350171 1.770933 2.16037 2.65031 3.01228 4.2208 0.258213 0.692417 1.345030 1.761310 2.14479 2.62449 2.97684 4.1405 0.257885 0.691197 1.340606 1.753050 2.13145 2.60248 2.94671 4.0728 0.257599 0.690132 1.336757 1.745884 2.11991 2.58349 2.92078 4.0150 0.257347 0.689195 1.333379 1.739607 2.10982 2.56693 2.89823 3.9651 0.257123 0.688364 1.330391 1.734064 2.10092 2.55238 2.87844 3.9216 19 0.256923 0.687621 1.327728 1.729133 2.09302 2.53948 2.86093 3.8834 20 0.256743 0.686954 1.325341 1.724718 2.08596 2.52798 2.84534 3.8495 21 0.256580 0.686352 1.323188 1.720743 2.07961 2.51765 2.83136 3.8193 0.256432 0.685805 1.321237 1.717144 2.07387 2.50832 2.81876 3.7921 0.256297 0.685306 1.319460 1.713872 2.06866 2.49987 2.80734 3.7676 0.256173 0.684850 1.317836 1.710882 2.06390 2.49216 2.79694 3.7454 0.256060 0.684430 1.316345 1.708141 2.05954 2.48511 2.78744 3.7251 0.255955 0.684043 1.314972 1.705618 2.05553 2.47863 2.77871 3.7066 0.255858 0.683685 1.313703 1.703288 2.05183 2.47266 2.77068 3.6896 0.255768 0.683353 1.312527 1.701131 2.04841 2.46714 2.76326 0.255684 0.683044 1.311434 1.699127 2.04523 2.46202 2.75639 30 0.255605 0.682756 1.310415 1.697261 2.04227 2.45726 2.75000 Large 0.253347 0.674490 1.281552 1.644854 1.95996 2.32635 2.57583 3.2905 22 27 28 3.6739 29 3.6594 3.6460 to.0005 To complicate matters, statisticians list confidence intervals backwards by using a concept called a level of significance, or a where: a = 1-%c For instance, the level of significance for a 90% confidence level would be a=0.1. The columns of the t-tables are arranged for various critical points of this a value (e.g. 10%, 5%, 1% etc). The corresponding t-values for various degrees of freedom at these significance levels are then listed as elements of the table. Table 1 Single sided Student's t distribution table listing to2 versus v. (probability higher than the critical t-value. Example: the probability that we will have a value exceeding the critical t- value of 2.228 for a degree of freedom of 10 is 0.025) to.40 to.05 ta2 D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 to.25 23 24 25 26 to.10 to.025 to.01 to.005 5.4079 5.0413 0.324920 1.000000 3.077684 6.313752 12.70620 31.82052 63.65674 636.6192 0.288675 0.816497 1.885618 2.919986 4.30265 6.96456 9.92484 31.5991 0.276671 0.764892 1.637744 2.353363 3.18245 4.54070 5.84091 12.9240 0.270722 0.740697 1.533206 2.131847 2.77645 3.74695 4.60409 8.6103 0.267181 0.726687 1.475884 2.015048 2.57058 3.36493 4.03214 6.8688 0.264835 0.717558 1.439756 1.943180 2.44691 3.14267 3.70743 5.9588 0.263167 0.711142 1.414924 1.894579 2.36462 2.99795 3.49948 0.261921 0.706387 1.396815 1.859548 2.30600 2.89646 3.35539 0.260955 0.702722 1.383029 1.833113 2.26216 2.82144 3.24984 4.7809 0.260185 0.699812 1.372184 1.812461 2.22814 2.76377 3.16927 4.5869 0.259556 0.697445 1.363430 1.795885 2.20099 2.71808 3.10581 4.4370 0.259033 0.695483 1.356217 1.782288 2.17881 2.68100 3.05454 4.3178 0.258591 0.693829 1.350171 1.770933 2.16037 2.65031 3.01228 4.2208 0.258213 0.692417 1.345030 1.761310 2.14479 2.62449 2.97684 4.1405 0.257885 0.691197 1.340606 1.753050 2.13145 2.60248 2.94671 4.0728 0.257599 0.690132 1.336757 1.745884 2.11991 2.58349 2.92078 4.0150 0.257347 0.689195 1.333379 1.739607 2.10982 2.56693 2.89823 3.9651 0.257123 0.688364 1.330391 1.734064 2.10092 2.55238 2.87844 3.9216 19 0.256923 0.687621 1.327728 1.729133 2.09302 2.53948 2.86093 3.8834 20 0.256743 0.686954 1.325341 1.724718 2.08596 2.52798 2.84534 3.8495 21 0.256580 0.686352 1.323188 1.720743 2.07961 2.51765 2.83136 3.8193 0.256432 0.685805 1.321237 1.717144 2.07387 2.50832 2.81876 3.7921 0.256297 0.685306 1.319460 1.713872 2.06866 2.49987 2.80734 3.7676 0.256173 0.684850 1.317836 1.710882 2.06390 2.49216 2.79694 3.7454 0.256060 0.684430 1.316345 1.708141 2.05954 2.48511 2.78744 3.7251 0.255955 0.684043 1.314972 1.705618 2.05553 2.47863 2.77871 3.7066 0.255858 0.683685 1.313703 1.703288 2.05183 2.47266 2.77068 3.6896 0.255768 0.683353 1.312527 1.701131 2.04841 2.46714 2.76326 0.255684 0.683044 1.311434 1.699127 2.04523 2.46202 2.75639 30 0.255605 0.682756 1.310415 1.697261 2.04227 2.45726 2.75000 Large 0.253347 0.674490 1.281552 1.644854 1.95996 2.32635 2.57583 3.2905 22 27 28 3.6739 29 3.6594 3.6460 to.0005