8. [-I9.61 Points] DEVORESTATB 5.4.0535. Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.5. (Round your answers to four decimal places.) IA use SA T (a) If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 8 pins is at least 51? (b) What is the (approximate) probability that the sample mean hardness for a random sample of 41 pins is at least 51? You may need to use the appropriate table in the Appendix of Tables to answer this question. 9. [-I9.61 Points] DEVORESTAT9 7.1.005. Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.77. (a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 25 specimens from the seam was 4.85. (Round your answers to two decimal places.) ( , I) (b) Compute a 98% CI for true average porosity of another seam based on 10 specimens with a sample average porosity of 4.56. (Round your answers to two decimal places.) ( , l) (c) How large a sample size is necessary if the width of the 95% interval is to be 0.42? (Round your answer up to the nearest whole number.) specimens (d) What sample size is necessary to estimate true average porosity to within 0.23 with 99% confidence? (Round your answer up to the nearest whole number.) specimens You may need to use the appropriate table in the Appendix of Tables to answer this