You know that variance is not standard deviation, so how is this possible? It is because the square root of one is one: √ 1 = 1 . Never forget that standard deviation is the square root of the value of the variance. This exception is only true for the standard normal distribution because “one” is a special number. Suppose you are given a question involving a certain normal random variable, say X , then you have to transform this variable X to fi t into the mould of the standard normal distribution, where it is customary to call the standardized random variable Z . The formula you use for this is Z = x − μ σ . A few pointers if you use the normal table: • Make sure to remember that the value in the denominator of this fraction is σ and not σ 2 . Read the information carefully and make sure whether you were given standard deviation or variance. If variance was given, fi rst fi nd the square root of that numerical value for the formula. • If you calculate P ( a < X < b ) , P ( a < Z < b ) , P ( Z < b ) , or P ( a < Z ) , you have to determine a probability , so make sure that your answer lies in the interval [0 , 1]! • Recall that the area under the normal curve is considered to be 1 . 0 . So, if the total area from −∞ to + ∞ is equal to 1 . 0 , then the area from −∞ to 0 is equal to 0 . 5 and the area from 0 to + ∞ is also equal to 0 . 5 .
142 • Never hesitate, for each question, to quickly make a small rough sketch of a normal curve (no art work!) and shade the area applicable for that question. That is very helpful and even some leturers do that!
143 STA1501/1 ___________________________________________________________________ Activity 8.2 Question 1 Given that Z is a standard normal random variable, P ( − 1 . 0 ≤ Z ≤ 1 . 5) is (1) 0 . 7745 (2) 0 . 8413 (3) 0 . 0919 (4) 0 . 9332 (5) 0 . 0994 Question 2 If Z is a standard normal random variable, then P ( − 1 . 75 < Z < − 1 . 25) is (1) 0 . 1056 (2) 0 . 0655 (3) 0 . 0401 (4) 0 . 8543 (5) − 0 . 0655 Question 3 Given that X is a normal variable, which of the following statements is/are true? (1) The variable X + 5 is also normally distributed. (2) The variable X − 5 is also normally distributed. (3) The variable 5 X is also normally distributed. (4) None of the above. (5) All of the above. Question 4 Given that the random variable X is normally distributed with a mean of 80 and a variance of 100 , P (85 < X < 90) is (1) 0 . 5328 (2) 0 . 2620 (3) 0 . 1915 (4) 0 . 1498 (5) 0 . 0199 ___________________________________________________________________
144 Finding values of Z In the questions discussed so far, you had to determine a probabilty for a given value of the random variable, but this can be turned around. A speci fi c probability is given and it is your task to fi nd the corresponding random variable. For this type of question, the same normal Table 3 is used, but in a “reverse” manner. A few pointers: • Do not hesitate, for each question, to make a small rough sketch of a normal curve and shade the given area. That gives you an idea of where the variable can be expected to fall. • Read the question very carefully. The interpretaion of > is quite different from < . Make sure your little sketch is correct!
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