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Chapter 6: Problem 4

Calculate the area under the standard normal curve between these values: a. \(z=-1.4\) and \(z=1.4\) b. \(z=-3.0\) and \(z=3.0\)

Short Answer

Expert verified
Question: Calculate the area under the standard normal curve between the given z-values for both cases (a) and (b). a) z = -1.4 and z = 1.4 b) z = -3.0 and z = 3.0 Answer: a) Approximately 0.8384 b) Approximately 0.9974

Step by step solution

01

Recall the properties of a standard normal curve

The standard normal curve, often represented by \(N(0, 1)\), represents a normal distribution with a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1. Typically, you will use a z-table to find the cumulative probability of a given \(z\)-value or look up the probability in statistical software.
02

Determine the area under the curve for case a

Using the z-table or statistical software, you need to find the cumulative probabilities for \(z = -1.4\) and \(z = 1.4\) and then subtract the smaller probability from the larger probability to determine the area between the two points. 1. Find the cumulative probability for \(z = -1.4\). In this case, the probability is approximately \(0.0808\). 2. Find the cumulative probability for \(z = 1.4\). In this case, the probability is approximately \(0.9192\). 3. Calculate the area between the two points: \(0.9192 - 0.0808 = 0.8384\). Therefore, the area under the standard normal curve between \(z=-1.4\) and \(z=1.4\) is approximately \(0.8384\).
03

Determine the area under the curve for case b

As in Step 2, use the z-table or statistical software to find the cumulative probabilities for \(z = -3.0\) and \(z = 3.0\), and then subtract the smaller probability from the larger probability to determine the area between the two points. 1. Find the cumulative probability for \(z = -3.0\). In this case, the probability is approximately \(0.0013\). 2. Find the cumulative probability for \(z = 3.0\). In this case, the probability is approximately \(0.9987\). 3. Calculate the area between the two points: \(0.9987 - 0.0013 = 0.9974\). Therefore, the area under the standard normal curve between \(z=-3.0\) and \(z=3.0\) is approximately \(0.9974\).

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