6554 19 chapter 6 b cfind p x 360

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Unformatted text preview: variable Z. This is done as follows: ⎛a−μ X−μ b−μ⎞ < < P(a < X < b) = P⎜ ⎟ σ σ⎠ ⎝σ b−μ⎞ ⎛a−μ = P⎜ <Z< ⎟ σ⎠ ⎝σ ⎛b−μ⎞ ⎛a−μ⎞ = F⎜ ⎟ − F⎜ ⎟ ⎝σ⎠ ⎝σ⎠ Appendix Table 1 of the textbook can now be used to look‐up the cumulative probabilities for the standard normal distribution. 17 Chapter 6 Example: Exercise 6.22, page 208. Let the continuous random variable X be the amount of money spent on textbooks by a student in September of the academic year. It is known that: X ~ N(μ , σ 2 ) with μ = $380 and σ = $50 Questions and Answers (a) Find P( X < 400) . This gives the probability that a randomly chosen student will spend less than $400 on textbooks in September. First state the probability as a probability about the standard normal variable Z: ⎛ X − μ 400 − μ ⎞ P(X < 400) = P⎜ < ⎟ σ⎠ ⎝σ 400 − 380 ⎞ ⎛ = P⎜ Z < ⎟ 50 ⎝ ⎠ = P(Z < 0.4) = F(0.4) Now look‐up the answer in Appendix Table 1 of the textbook. The table gives: F(0.4) = 0.6554 Therefore, P( X < 400) = 0.6554 18 Chapter 6 A graph gives a helpful illustration of the use of the statistical tables for this problem. PDF of Z f(z) Area = F(0.4) = 0.6554 0 0.4 z Now check the answer with Microsoft Excel by selecting Insert Function: NORMDIST(x, μ X , σ X , cumulative...
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