 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that there is a production lot of 2,520 fence pliers with hammer head and 30 pieces are packed per master carton. There are 84 master cartons in all. It is assumed that the cutters will be able to cut a 6 gauge that is 4 mm diameter Steel wire if their hardness is within 50 to 55 HRC Rockwell. What inference can be drawn for the entire production lot from the randomly selected samples? We know that in normal distribution about 68% of the data or area lies between X bar minus sigma and X bar plus sigma about 95% of data or area lies between X bar minus 2 sigma and X bar plus 2 sigma and about 99% of data or area lies between X bar minus 3 sigma and X bar plus 3 sigma. Also we know that proportion means a part considered in relation to a whole and population Proportion is the proportion of individuals in a population sharing a certain trait and it is denoted by P. Similarly sample proportion is the proportion of individuals in a sample sharing a certain trait and is denoted by P cap. Sample proportion is the estimate of sample proportion with X successes in N trials that is sample proportion P cap is equal to X upon N. Also we know that formula to find population proportion P for 95% confidence interval is given by P cap minus 2 into square root of P cap into 1 minus P cap the whole whole upon N is less than P is less than P cap plus 2 into square root of P cap into 1 minus P cap the whole whole upon N where P is the population proportion P cap is sample proportion and N is the size of the sample with this key idea we shall proceed to the solution. We are given a lot of 2520 fence pliers and 30 pieces are packed per master carton there are 84 master cartons in all so we randomly select say 40 pliers from different cartons and note the hardness of their cutters. Values came out to be as follows here we have hardness values in Rockwell from 40 randomly selected pliers so here N is equal to 40 that is the sample size N is given by 40 now we shall calculate sample proportion of the 40 pliers now we are given in the question that the cutters will be able to cut a 6 gauge steel wire if their hardness is within 50 to 55 Rockwell and from our sample we know that there are 1 2 3 4 5 6 and 7 pliers out of this range and the rest 33 pliers have hardness values which are within the specified range and from the key idea we know that sample proportion is the estimate of sample proportion with X successes in N trials and sample proportion P cap is given by X upon N here sample proportion denoted by P cap will be given by X upon N that is X is the number of successes and here we have excess 33 as 33 pliers are within the specified range and N is the sample size which is given by 40 so sample proportion P cap is given by 33 upon 40 which is equal to 0.825 we repeat the same process 9 times and calculate sample proportion for each of the sample lots reason for taking 10 lots is to get 10 values of sample proportion thereby enabling us to safely assume that the values obtained form a normal curve this also conveys that we have to inspect approximately 15% of the production lot the values of sample proportion obtained from 9 other sample lots are as follows 0.8, 0.825, 0.850, 0.825, 0.875, 0.850, 0.9, 0.875 and 0.925 now we shall take out the mean of all these 10 sample proportion values and it is equal to sum of all these values divided by 10 and this is given by 8.55 upon 10 and it is equal to 0.855 from the key idea we know that formula to find population proportion P for 95% confidence interval is given by P cap minus 2 into square root of P cap into 1 minus P cap the whole whole upon N is less than P is less than P cap plus 2 into square root of P cap into 1 minus P cap the whole whole upon N where P is the population proportion P cap is the sample proportion and N is the size of the sample we know that in normal distribution 95% of the data lies between X bar minus 2 sigma and X bar plus 2 sigma so we put the values of P cap and N in this formula to make an inference about population proportion and here we have calculated the value of P cap as 0.855 and N is 40 therefore we get 0.855 minus 2 into square root of 0.855 into 1 minus 0.855 the whole upon 40 is less than P is less than 0.855 plus 2 into square root of 0.855 into 1 minus 0.855 the whole whole upon 40 which implies that 0.855 minus 2 into square root of 0.855 into 1 minus 0.855 that is 0.145 whole upon 40 is less than P is less than 0.855 plus 2 into square root of 0.855 into 1 minus 0.855 that is 0.145 whole upon 40 this further implies that 0.855 minus 2 into square root of 0.855 into 0.145 that is 0.124 upon 40 is less than P is less than 0.855 plus 2 into square root of 0.855 into 0.145 that is 0.124 upon 40 and this implies that 0.855 minus 2 into square root of 0.124 upon 40 that is 0.0031 is less than P is less than 0.855 plus 2 into square root of 0.124 upon 40 that is 0.0031 which further implies 0.855 minus 2 into square root of 0.0031 that is 0.056 is less than P is less than 0.855 plus 2 into square root of 0.0031 that is 0.056 which implies that 0.855 minus 2 into 0.056 that is 0.112 is less than P is less than 0.855 plus 2 into 0.056 that is 0.112 this further implies that 0.855 minus of 0.112 is 0.743 is less than P is less than 0.855 plus 0.112 that is 0.967 thus we have got 0.743 is less than P is less than 0.967 now we need to work out the actual number of pieces which are fine this is arrived at by multiplying the two end values of the population proportion obtained above by the size of the production lot so we get 0.743 into 2520 which is the total size of the production lot and this is equal to 1872.36 which is approximately equal to 1872 which is equal to 74.3% of the pliers and we multiply 0.967 with 2520 and we get 2436.84 which is approximately equal to 2437 which is equal to 96.67% of the pliers that we can say that the population proportion lies within the range of 0.743 to 0.967 in the 95% confidence interval it means we are 95% confident that 1872 to 2437 pliers are of fine quality it can be difficult to take a decision regarding accepting or rejecting a production lot based on justice analysis as commitments, risk, validity of assumptions and other subjective factors are involved quantitative analysis by itself is insufficient for decision making qualitative analysis is also required however based on the quantitative analysis done here we are able to derive comfort in being 95% confident that 1872 to 2437 pliers are of fine quality which is the required answer this completes our session hope you enjoyed this session