We need to first convert the probability P(x < 96 ) to a corresponding probability involving a z value. Recall we previously found that z = 3 when x = 96 find P(z < )
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(a) P(0 ≤ Z ≤ 2.96) Recall that a standard normal variable, Z, has mean μ = 0 and standard deviation σ = 1. The cumulative density function (cdf) for Z, denoted by Φ(z) = P(Z ≤ z), is the area under the standard normal curve to the left of z, as shown below. A plot of the normal probability curve has a horizontal axis with values from -3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the left of 2.96 is shaded. A table of select Φ(z) values is given in Table A.3 of the appendix. We want to calculate the probability P(0 ≤ Z ≤ 2.96). This is the area under the standard normal curve between z = 0 and z = 2.96, shown in the graph below. A plot of the normal probability curve has a horizontal axis with values from -3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve between 0 and 2.96 is shaded. Recall that if X is a continuous variable with cdf F(x), then for any two numbers a and b with a < b, we have the following property. P(a ≤ X ≤ b) = P(X ≤ b) - P(X < a) = F(b) - F(a) That is, the area between a and b can be calculated by finding the entire area to the left of b and subtracting the area to the left of a. Applying this property for the variable Z with cdf Φ(z), we have P(a ≤ Z ≤ b) = Φ(b) - Φ(a). Thus, we can calculate the area of the shaded region, P(0 ≤ Z ≤ 2.96), by finding the area to the left z = b = 2.96, and subtracting the area to the left of z = a = 0. That is, we can write P(0 ≤ Z ≤ 2.96) = Φ(2.96) - Φ(0).
Adi S.
Use a z-score chart to calculate P ( Z < - 0.96 )
Pritesh R.
Use Table II (area under the standard normal curve) in Appendix A to find the probability. P(92≤x≤116) Use Table II (area under the standard normal curve) in Appendix A to find the probability. P(92≤x≤96) Use Table II (area under the standard normal curve) in Appendix A to find the probability. P(76≤x≤124)
Ivan K.
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