8.8 APPLICATIONS OF THE NORMAL CURVE


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1 8.8 APPLICATIONS OF THE NORMAL CURVE If a set of real data, such as test scores or weights of persons or things, can be assumed to be generated by a normal probability model, the table of standard normalcurve areas can be applied to solve the problem even when the data are not standard normal by standardiing the quantities involved in the problem. Example 8.5 The lifetime of a certain type of battery has been found to be normally distributed with a mean of 200 hours and a standard deviation of 15 hours (this sort of information could be gathered, for example, by keeping records at the factory on the lifetimes of samples of batteries over a long period of time). What proportion of these batteries can be expected to last less than 220 hours? Probabilities can be thought of as proportions, so this is a probability problem. The assumed distribution of battery lifetimes is shown in Figure The shaded region shows the batteries that have a lifetime of less than 220 hours. We can calculate this area using the table of normalcurve areas if we know the statistic value corresponding to 220 hours Thus p(lifetime 220) p( 1. 33), where 1.33 is 220 standardied.
2 Hours Figure 8.20 p(battery life 220 hours). We now know the value of that corresponds to 220 hours. Using this value, we can get the area of the shaded region in Figure 8.20 from the table of normalcurve areas (again, assuming that the battery lives are generated by a normal probability model). That is, we can find p ( 1. 33) From the table, we find the area to be , so the proportion of batteries with lifetimes shorter than 220 hours (which corresponds to a of 1.33) is expected to be 0.91 (rounded). That is, about 91% of the batteries can be expected to last less than 220 hours. Example 8.6 What proportion of the batteries from Example 8.5 can be expected to last more than 220 hours? Again, assuming the battery lifetimes to be distributed normally, we have the graph shown in Figure The shaded region represents the proportion of batteries with lifetimes greater than 220 hours. Since the corresponding to 220 hours is 1.33, we want to find p ( 1. 33) We know from the table that p ( 1. 33) 0. 91
3 Hours Figure 8.21 p(battery life 220 hours). Once again we use the fact that p(event) 1 p(complement of the event). So p ( 1. 33) We expect about 0.09 (or 9%) of the batteries to last more than 220 hours. Example 8.7 The SPWEHIQS Club (Society of People with Extremely High IQs) requires people to take an intelligence test as a condition for joining the club and restricts membership to the top 5% as measured by this test. Suppose the scores on the intelligence test have a mean of 100 and a standard deviation of 12 and are normally distributed for very large groups of people. What is the lowest score on this test that would be acceptable for admission to SPWEHIQS? Figure 8.22 shows the distribution of the scores of the intelligence test. The figure indicates that 5%, or a proportion of 0.05, of the total area is in the shaded region under the curve. According to the normalcurve table (Table E), this area corresponds to a statistic of A statistic here is computed as IQ Thus a statistic can be changed to an intelligence test score by a scaling factor of 12 (the standard deviation of the test scores) and a centering factor of 100 (the mean of the test scores). So the test score corresponding to a statistic (score) of 1.65 is IQ 12(1. 65)
4 Area = IQ = Figure 8.22 p (IQ?) Rounding this score to 120, we find that a score of 120 would be the lowest acceptable for admission to the club. Example 8.8 A certain insect species has a mean length of 1.2 centimeters and a standard deviation of 0.12 centimeters. If there are estimated to be 1000 of these insects in a terrarium, how many would be expected to be less than 1 centimeter in length? Assume that the lengths are normally distributed. Figure 8.23 is a sketch of the distribution of the insect lengths. The shaded part of the graph begins at the 1centimeter mark and includes the region to the left of this point (since the problem specifies insects having lengths less than 1 centimeter). The statistic corresponding to 1 centimeter is found by the transformation According to Table E, the area to the left of is So a proportion of (4.75%) of the 1000 insects are expected to be less than 1 centimeter long. That is, we expect to be less than 1 centimeter in length insects
5 Area = Length = Figure 8.23 p(insect length 1 cm). SECTION 8.8 EXERCISES Suppose 30 million boxes of bananas are packed a year. Given this packing plant For each of the following problems, if the exact rule and the assumption about the distrivalue is not available in the table, use the closest bution of box weights upon arrival, how many one. boxes would be expected to weigh less than 1. The heights of a group of male students follow 40 pounds upon arrival? ( Note: Tables tell us the normal distribution with mean 70 inches that for a standard normal, p( 4. 00) and standard deviation 3.1 inches ) a. What percentage of the students would 3. A car manufacturer is producing a piston you expect to be shorter than 68 inches? for its engines. The piston is supposed to b. What percentage of the students would have a diameter of 5.3 inches, but because of you expect to be taller than 73.5 inches? variability, the diameter of a piston actually c. What height are 31% of the students follows a normal distribution with a mean of shorter than, and what height are 69% of 5.3 and a standard deviation of If a piston them taller than? is more than inch away from the needed 2. To help ensure that boxes of its bananas weigh sie of 5.3 inches, the piston is rejected. at least 40 pounds upon arrival at their destiexpect to be too large? a. What percentage of the pistons would you nation, a packing plant might adopt this rule: Pack boxes to have a weight of 41.5 pounds of b. What percentage of the pistons would you bananas with a maximum permissible range expect to be too small? Hint: Recall that a of 3 ounces above or below 41 pounds, 8 probability is converted to a percentage by ounces (that is, pack boxes to have at least moving the decimal point two places right. 41 pounds, 5 ounces, but no more than Recall the situation presented in Exercise 3. pounds, 11 ounces, of bananas). With this rule Suppose instead, because of a mechanical and the shrinkage in travel, the distribution of problem with the machine producing the pis box weights upon arrival may be assumed to tons, the pistons have diameters that follow be approximately normal with a mean of 41 a normal distribution with a mean of 5.29 pounds and a standard deviation of 4 ounces. inches. The standard deviation is still 0.01.
6 The specifications require the diameter of the piston to be 5.3 plus or minus a. What percentage of the pistons would you expect to be too large in this situation? b. What percentage of the pistons would you expect to be too small? 5. A set of final exam scores has a mean of 52 and a standard deviation of 6. The scores are normally distributed. If a teacher wants to assign a grade of A to the top 15% of the scores, what score should be the lowest A? If the bottom 15% are to be Fs, what test score should be the highest F? 6. A pipette is a precise instrument used in science to dispense an exact amount of liquid. The pipette is carefully calibrated, but, as with any measurement, it will not dispense the same amount each time. The error in amount is well described by the normal distribution. Suppose a scientist is using a pipette known to dispense amounts with a standard deviation of 0.05 microliter (a microliter is a millionth of a liter). The scientist is able to set the pipette to dispense varying amounts of liquid. a. If the pipette is set to dispense 200 microliters, what percentage of the time will it dispense greater than microliters? b. If the pipette is set to dispense 175 mi croliters, what percentage of the time will it dispense less than microliters of liquid? c. The scientist wanted to set the pipette to dispense 200 microliters, but she acciden tally set it to 199. Is there any chance that the pipette will dispense as much as 200 microliters? For additional exercises, see page 727.
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