Suppose X is a random variable with a normal distribution with mean mu and standard deviation sigma. What is the value of z that makes this a true statement? ( Find Z that satisfies this equation )

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−
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Question

Suppose X is a random variable with a normal distribution with mean mu and standard deviation sigma. What is the value of  z that makes this a true statement? ( Find Z that satisfies this equation )

𝑃
(
−
𝑧
<
𝑋
−
𝜇
𝜎
<
𝑧
)
=
 .96

Draw picture and label it.

BlackTom AI · Accepted Answer

We start by restating the problem in standard form. Let Z be a standard normal random variable, Z ~ N(0,1). The quantity (X − μ)/σ has the standard normal distribution, so the probability statement becomes P(−z < Z < z) = 0.96.

Next, recognize the symmetry of the standard normal distribution. The probability of Z lying between −z and z can be written using the CDF Φ as P(−z < Z < z) = Φ(z) − Φ(−z). Because Φ(−z) = 1 − Φ(z) for a standard normal, this expression simplifies to Φ(z) − (1 − Φ(z)) = 2Φ(z) − 1.

Set up the equation with the given probability: 2Φ(z) − 1 = 0.96. Solving for Φ(z) gives Φ(z) = (0.96 + 1)/2 = 0.98.

Now we need the z-value corresponding to a standard normal CDF of 0.98. This is the inverse CDF, often denoted Φ⁻¹(0.98). Numerically, Φ⁻¹(0.98) is approximately 2.05 (more precisely about 2.0537), which is the z-value that yields a 96% central probability around the mean.

To visualize, imagine the standard normal curve with the interval [−z, z] centered at 0 containing 96% of the area under the curve. Since 0.98 is near the upper tail, z falls around the 98th percentile, which is just above 2 standard deviations from the mean.

If you were to sketch: draw the bell-shaped curve, mark z to the right of zero, shade the region between −z and z, and note that this shaded area should cover 0.96 of the total area. The corresponding z-axis value where Φ(z) = 0.98 is about 2.05.

BTRY3010/BTRY5010/STSCI2200/STSCI5200 Lecture#21 (Pages 102-103)

Suppose X is a random variable with a normal distribution with mean mu and standard deviation sigma. What is the value of z that makes this a true statement? ( Find Z that satisfies this equation ) 𝑃 ( − 𝑧 < 𝑋 − 𝜇 𝜎 < 𝑧 ) = .96 Draw picture and label it.

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