Questions: Find the area under the standard normal curve between z=0.02 and z=2.69. Round your answer to four decimal places, if necessary.
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Lesson: 6.2 The Standard Normal Distribut...
Question 8 of 12 . Step 1 of 1
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Find the area under the standard normal curve between $z=0.02$ and $z=2.69$. Round your answer to four decimal places, if necessary.
Answer
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Normal Table $-\infty$ to $-z$
Normal Table $-\infty$ to $z$
Solution
Solution Steps
Step 1: Define the Problem
We need to find the area under the standard normal curve between \( z = 0.02 \) and \( z = 2.69 \). This area can be expressed as the difference between the cumulative distribution function (CDF) values at these two z-scores.
Step 2: Calculate the CDF Values
The area can be calculated using the formula:
\[
P = \Phi(Z_{end}) - \Phi(Z_{start})
\]
where \( \Phi(z) \) is the cumulative distribution function of the standard normal distribution.
Step 3: Substitute the Z-scores
Substituting the values into the equation:
\[
P = \Phi(2.69) - \Phi(0.02)
\]
Step 4: Evaluate the CDF Values
From the calculations, we find:
\[
P = 0.4884
\]
Final Answer
The area under the standard normal curve between \( z = 0.02 \) and \( z = 2.69 \) is given by:
\[
\boxed{P = 0.4884}
\]