CONSERVACIÓN DE LA BIODIVERSIDAD
3.6 ASPECTOS METODOLÓGICOS
3.6.3 Superposición de la capa de ANAPRO y la de Sistemas Ecológicos
Step 3: Input a value from to to select the probability distribution calculation we want perform.
Example 23
Find the following probabilities, by using Casio Scientific Calculator fx-570W or fx-
570MS. (a) P(Z 0.75). (b) P(Z 0.95). (c) P(Z 1.42). (d) P(Z 1.43). (e) P(|Z|1.23). (f) P(|Z |0.99). (g) P(0.67Z 1.34). (h) P(1.74Z 0.75). (i) P(1.54Z 2.32). P( Q( R( →t 1 2 3 4 1 3 P(t) Q(t) R(t)
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Answer Example 23
(a) SHIFT DISTR 3
0.75 ) = 0.2266
(b) SHIFT DISTR 1 0.95 ) = 0.8289
(c) SHIFT DISTR 3 1.42 ) = 0.0778
(d) SHIFT DISTR 1 1.43 ) = 0.9236
(e) SHIFT DISTR 3 1.23 ) X 2 = 0.2187
(f) SHIFT DISTR 2 0.99 ) X 2 = 0.6778
(g) SHIFT DISTR 3 0.67 )
- SHIFT DISTR 3 1.34 ) = 0.1613
(h) SHIFT DISTR 3 0.75 )
- SHIFT DISTR 3 1.74 ) = 0.1901
(i) 1 - SHIFT DISTR 3
1.54 )
- SHIFT DISTR 3 2.32 ) = 0.9281
Example 24
Let exam marks for 200 students are normally distributed with 50 and 2 16. Find the number of student that passed the exam (the marks are 40).
Answer Example 25
Let X be the exam marks, X ~ N
50,16
.
2.5
16 ` 50 40 40 P Z P Z X P 1P
Z 2.5
10.0062 0.993894
Exercise 3.5
1. If the height of 300 students is normally distributed with mean 68 in and standard deviation 0.0025 in, how many students have height greater than 72 in?
2. Studies show that gasoline use for compact cars sold in the Batu Pahat is normally distributed, with a mean of 25.5 miles per gallon (mpg) and a standard deviation of 4.5 mpg. What percentage of compacts gets 30 mpg or more?
3. A study of lease rates for a selection of 2006 cars revealed that the average monthly rate for a vehicle was RM 220.67. The standard deviation was RM 59.63. Assume that the rates follow a normal distribution.
(a) What is the probability that a random car will lease for less than RM 150?
(b) Find the probability that a random car will lease for more than RM 350?
4. Based on data from ACT in 2007, the average science reasoning test score was 20.9, with a standard deviation of 4.6. Assuming that the scores are normally distributed:
(a) Find the probability that a randomly selected student has a science reasoning ACT score of least 25.
(b) Find the probability that a randomly selected student has a science reasoning ACT score between 20 and 26.
5. The average annual charges per credit card in 2006 were RM 9600 according from one research. Assuming that the annual charges per card are approximately normally distributed with a standard deviation of RM 2100, what is the probability that a credit card customer’s annual charges are
(a) less than RM 4000
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6. The grades on an examination whose mean is 525 and whose standard deviation is 80 are normally distributed. What is the probability that the person will score the examination below than 350 ?
7. For a car traveling 30 miles per hour (mph), the distance required to brake to a stop is normally distributed with a mean of 50 feet and a standard deviation of 8 feet. Suppose you are traveling in a residential area and a car abruptly into your path at a distance of 60 feet. What is the probability that you will
(a) brake a stop within 40 feet or less ?
(b) Brake a stop between 40 feet and 45 feet ?
8. Suppose that you must establish regulations concerning the maximum number of people who can occupy an elevator. A study of elevator occupancies indicates that if eight people occupy the elevator, the probability distribution of the total weight of the eight people has a mean equal to 1200 pounds and a standard deviation of 99 pounds. What is the probability that the total weight of eight people exceeds 1300 pounds ?
9. Suppose that the unsupported stem diameters at the base of a particular species of sunflower plant have a normal distribution with an average diameter of 35 millimeter and a standard deviation of 3 millimeter.
(a) What is the probability that a sunflower plant will have a basal diameter of more than 40 mm ?
(b) What is the probability that a sunflower plant will have a basal diameter between 30 mm and 40 mm ?
10. The meat department at a Parit Raja supermarket specifically prepares its “1 kg” packages of ground beef so that there will be a variety of weights, some slightly more and more slightly less than 1 kg. Suppose it is normally distributed with a mean of 1 kg and a standard deviation of 0.15 kg. What is
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the probability of the packages will weight between 0.95 kg and 1.05 kg ?
11. The discharge of suspended solids from a phosphate mine is normally distributed, with a mean daily discharge of 27 milligrams per liter (mg/l) and a standard deviation of 14 mg/l. What is the probability if the daily discharge exceed 50 mg/l ?
12. Suppose the individual 2008 January prime interest rate forecasts of economic analysts are approximately normally distributed with the mean equal to 8.5 and a standard deviation equal to 0.2. If a single analyst is randomly selected from among this group, what is the probability that the analyst’s forecast of the prime rate will take on these values ?
(a) exceed 8.75 (b) less than 8.375
13. Suppose the numbers of a particular type of bacteria in samples of 1 milliliter of drinking water tend to be approximately normally distributed, with a mean of 85 and a standard deviation of 9. What is the probability it will more than 100 bacteria?
14. How does the income tax bureau decide on the percentage of income tax returns to audit for each state? Suppose it is approximately normally distributed with a mean 1.55% and a standard deviation 0.45%.
(a) What is the probability that a particular state will have more than 2.5% of its income tax returns audited ?
(b) What is the probability that a state will have less than 1% of its income tax returns audited ?
15. A study have been done to look at the amount of money spent at shopping complex between 4 pm and 6 pm on Sundays and had a normal distribution with a mean RM 85 and with a standard deviation of RM 20. A shopper is
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randomly selected on a Sunday between 4 pm and 6 pm and asked about their spending patterns.
(a) What is the probability that they spent more than RM 95 at the mall ? (b) What is the probability that they spent between RM 95 and RM 115 at
the mall ? Answer Exercise 2.5 1. 20 2. 15.87% 3. (a) 0.1170 (b) 0.0150 4. (a) 0.1867 (b) 0.4458 5. (a) 0.0038 (b) 0.5610 6. 0.01435 7. (a) 0.10565 (b) 0.16034 8. 0.1562 9. (a) 0.0475 (b) 0.90448 10. 0.26086 11. 0.05021 12. (a) 0.1056 (b) 0.2676 13. 0.04779 14. (a) 0.01738 (b) 0.11085 15. (a) 0.3085 (b) 0.2417
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2.6 Normal approximation to binomial distribution
It should not be surprise to learn that the normal distribution can be used to approximate binomial probabilities for cases in which n is large. Normal approximation should be used only when np5 and nq5. Figure (a) shows the probability for a binomial distribution. Figure (b) shows an approximation of binomial probabilities by normal probabilities.
Because binomial distribution is discrete and the normal distribution is continuous, then we must use the continuity correction (adding and subtracting 0.5). For example, to find the probability of 12 successes, we will use P
11.5 X 12.5
.Theory 14
Binomial Distribution Normal Distribution
(i) P(X = a) P(a – 0.5 < X < a + 0.5)
(ii) P(X a) P(X > a – 0.5)
(iii) P(X > a) P(X > a + 0.5)
(iv) P(X a) P(X < a + 0.5)
(v) P(X < a) P(X < a – 0.5)
(vi) P(a X b) P(a – 0.5 < X < b + 0.5)
(vii) P(a < X < b) P(a + 0.5 < X < b – 0.5)
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Theory 15
There are five steps to use normal approximation to binomial Distribution
Step 1 : Check whether the normal approximation can be used. Step 2 : Find the value of mean and standard deviation.