What is the approximate area of the unshaded region under …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

What is the approximate area of the unshaded region under the standard normal curve below? Use the portion of the standard normal table given to help answer the questions. See photo’s attached.

Short Answer

To find the area of the shaded region under the standard normal curve, calculate P(-2 ‚a§ z ‚a§ 1) using cumulative density values Œ¶(1) = 0.8413 and Œ¶(-2) = 0.0228, resulting in P(-2 ‚a§ z ‚a§ 1) = 0.8185. Consequently, the area of the unshaded region is approximately 0.18, found by subtracting the shaded area from the total area of 1.

Step-by-Step Solution

Step 1: Find the Area of the Shaded Region

To determine the area of the shaded region under the standard normal curve, start by calculating P(-2 ‚a§ z ‚a§ 1). This involves finding two cumulative density values: Œ¶(1) and Œ¶(-2). The area of the shaded region can be expressed as:

  • P(-2 ‚Äöa¬ß z ‚Äöa¬ß 1) = ≈í¬∂(1) – ≈í¬∂(-2)

Step 2: Calculate Cumulative Density Values

Use the standard normal distribution table to look up the cumulative densities. For Φ(1), the value is 0.8413. For Φ(-2), since Φ(2) is given as 0.9772, you can find Φ(-2) using the symmetry of the normal curve:

  • ≈í¬∂(-2) = 1 – ≈í¬∂(2) = 1 – 0.9772 = 0.0228

Step 3: Find the Area of the Unshaded Region

Now, you can calculate the area of the unshaded region by subtracting the shaded area from the total area of 1. Using the value from Step 1:

  • Area of unshaded region = 1 – P(-2 ‚Äöa¬ß z ‚Äöa¬ß 1)
  • Area of unshaded region = 1 – 0.8185 = 0.1815

The final result rounds to approximately 0.18, which is the area of the unshaded region.

Related Concepts

Cumulative Density Function

A function that describes the probability that a random variable takes a value less than or equal to a given value, commonly denoted as Ϧ(z) in the context of the standard normal distribution.

Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1, represented by the variable z. it is used as a reference for calculating probabilities related to normally distributed data.

Area Under The Curve

The total probability represented by the integral of the probability density function over a certain range, which in the case of normal distributions sums up to 1 across the entire range of values.

Table Of Contents
  1. What is the approximate area of the unshaded region under the standard normal curve below? Use the portion of the standard normal table given to help answer the questions. See photo’s attached.

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