Calculating Values from ProbabilitiesAs stated in the previous lesson, to find the probabilities of something normally distributed occurring, simply transform the data into a standard normal distribution, which has already been studied using the table below:Normal distributionThe transformation is given by the following formula:\[ z = \frac{x - \bar{x}}{\sigma} \]However, sometimes we want to know what happens with probability X. To calculate values from probabilities, we use the following steps:- Remember that percentages or probabilities are areas of the graph, not z values- Reading the table is reversed (from the area, one finds it)- Choose the correct side of the graph (the longer times are on the right side)- Apply the variation of the standardization formulaThe variation of the standardization formula consists of:\[ z = \frac{x - \bar{x}}{\sigma} \]\[ z*\sigma = x - \bar{x} \]\[ x = \bar{x} + z · \sigma \]In other words, with the standard deviation and the mean, any value can be calculated for a given probability, since Z is found from it.ExamplesConsider a factory where workers take an average of 75 minutes to complete a task, with a standard deviation of 6 minutes. How much time separates the fastest 10% from the slowest 10%?The fastest 10% are on the left side of the graph; if we analyze it, the Z for the slowest 10% has a value of: -1.28The slowest 10% are on the right side of the graph; if we analyze it, the z for the fastest 10% has a value of: 1.28Thus, calculate the time of the fastest 10%:\[ x = \bar{x} + z · \sigma \]\[ x = 75 - 1.28*6 = 67.32 \]And of the slowest 10%:\[ x = \bar{x} + z · \sigma \]\[ x = 75 + 1.28*6 = 82.68 \]The difference between the two times is:\[ \text{difference } = 82.68 - 67.32 = 15.36 \]Consider a factory where workers take an average of 75 minutes to complete a task, with a standard deviation of 6 minutes. What is the minimum time taken by the fastest 5%?The slowest 5% are on the right side of the graph; if we analyze it, the z for the fastest 5% has a value of: 1.65Thus, calculate the minimum time of the fastest 5%:\[ x = \bar{x} + z · \sigma \]\[ x = 75 + 1.65 · 6 = 84.9 \]Symmetry in the DistributionIt has already been discussed that the normal distribution is symmetric. This characteristic is often explored:- Richer or poorer- Faster or slower- Taller or shorterHere is a warning: in exercises where there is a duality, remember that each duality represents 50% of the distribution.ExamplesConsider a factory where workers take an average of 75 minutes to complete a task, with a standard deviation of 6 minutes. What time separates the 10% among the fastest from the 10% among the slowest?In this case, 10% among the fastest, which are 50% of the distribution, means 5% of the distribution on the right side. Meanwhile, 10% among the slowest, which are 50% of the distribution, means 5% of the distribution on the left side.The z for the fastest 5% has a value of: 1.65\[ x = \bar{x} + z · \sigma \]\[ x = 75 + 1.65 · 6 = 84.9 \]The z for the slowest 5% has a value of: -1.65\[ x = \bar{x} + z · \sigma \]\[ x = 75 - 1.65 · 6 = 65.1 \]The difference:\[ \text{difference } = 84.9 - 65.1 = 19.8 \]Exercises1) Suppose that the time required to serve customers in a telephone call center follows a normal distribution with a mean of 8 minutes and a standard deviation of 2 minutes. 25% of telephone calls require at least how much service time?2) Suppose that the time required to serve customers in a telephone call center follows a normal distribution with a mean of 8 minutes and a standard deviation of 2 minutes. 75% of telephone calls require at least how much service time?3) Suppose that the time required to serve customers in a telephone call center follows a normal distribution with a mean of 8 minutes and a standard deviation of 2 minutes. What is the probability that a service call lasts less than 5 minutes? And more than 9.5 minutes? And between 7 and 10 minutes?4) The distribution of weights of rabbits raised on a farm can very well be represented by a Normal distribution, with a mean of 5 kg and a standard deviation of 0.9 kg. A slaughterhouse will buy 5000 rabbits and intends to classify them according to weight as follows: the lightest 15% as small, the next 50% as medium, the next 20% as large, and the heaviest 15% as extra. What are the weight limits for each classification?5) An automatic soft drink filling machine is set so that the average volume of liquid in each bottle is 1000 cm³ and the standard deviation is 10 cm³. Assume that the volume follows a normal distribution. What is the percentage of bottles in which the volume of liquid is less than 990 cm³? What is the percentage of bottles in which the volume of liquid does not deviate from the mean by more than two standard deviations?6) A company produces televisions of 2 types, type A (standard) and type B (luxury), and guarantees a refund of the amount paid if any television presents a serious defect within six months. The time until the occurrence of a serious defect in the televisions has a normal distribution; for type A, with a mean of 10 months and a standard deviation of 2 months, and for type B, with a mean of 11 months and a standard deviation of 3 months. Type A and B televisions are produced with a profit of R\$1200 and R\$2100, respectively, and, in the event of a refund, with a loss of R\$2500 and R\$7000, respectively. Calculate the probabilities of a refund occurring for type A and type B televisions and their average profits.7) A car factory knows that the engines it manufactures have a normal lifespan with a mean of 150000 km and a standard deviation of 5000 km. What is the probability that a randomly chosen car manufactured by this company has an engine that lasts: less than 170000 km? Between 140000 km and 165000 km? If the factory replaces any engine whose lifespan is below the warranty, what should this warranty be so that the percentage of engines replaced is less than 0.2%?8) The mean of the internal diameters of a sample of 200 washers produced by a certain machine is 0.502 cm and the standard deviation is 0.005. The purpose for which these washers are manufactured allows a maximum tolerance, for the diameter, from 0.496 to 0.508 cm. If this is not met, the washers will be considered defective. Determine the percentage of defective washers produced by the machine, assuming that the diameters are normally distributed.9) Suppose that the measurements of electric current in a piece of wire follow the Normal distribution, with a mean of 10 milliamperes and a variance of 4 milliamperes. What is the probability that the measurement exceeds 13 milliamperes? What is the probability that the current measurement is between 9 and 11 milliamperes?10) Suppose that the measurements of electric current in a piece of wire follow the Normal distribution, with a mean of 10 milliamperes and a variance of 4 milliamperes. Determine the value for which the probability that a current measurement is below this value is 98%.11) The diameter of the main spindle of a hard drive follows the Normal distribution with a mean of 25.08 cm and a standard deviation of 0.05 cm. If the specifications for this spindle are 25.00 ± 0.15 cm, determine the percentage of units produced in compliance with the specifications.12) The concentration of a pollutant in water released by a factory has distribution N(8; 1.5). Mean 8 and variance 1.5. What is the chance that, on a given day, the concentration of the pollutant exceeds the regulatory limit of 10 ppm?Answer Key1) 6.7 minutes2) 9.34 minutes3) 6.68%; 22.66%; 53.28%;4) small < 4.1 kg < medium < 5.4 kg < large < 5.9 kg < extra5) 15.9%; 95%;6) Type A -> 2.28% and average profit of R\$1115.64; Type B -> 4.75% and average profit of R\$1667.757) 99.993%; 97.59%; warranty of 135650 km.8) 23%;9) 6.68%; 38.29%;10) 14.1 milliamperes;11) 91.92%12) 5.2%
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