Questions: Find the indicated area under the standard normal curve. To the left of z=-1.78 Click here to view page 1 of the standard normal table. The area to the left of z=-1.78 under the standard normal curve is (Round to four decimal places as needed.)

Find the indicated area under the standard normal curve.
To the left of z=-1.78
Click here to view page 1 of the standard normal table.

The area to the left of z=-1.78 under the standard normal curve is
(Round to four decimal places as needed.)

Transcript text: Find the indicated area under the standard normal curve. To the left of $z=-1.78$ Click here to view page 1 of the standard normal table. The area to the left of $z=-1.78$ under the standard normal curve is $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Define the Problem

We need to find the area under the standard normal curve to the left of $ z = -1.78 $. This area can be represented mathematically as $ P(Z < -1.78) $.

Step 2: Use the Cumulative Distribution Function

The area to the left of a given $ z $-score can be calculated using the cumulative distribution function (CDF) of the standard normal distribution, denoted as $ \Phi(z) $. Therefore, we can express the area as: \[ P(Z < -1.78) = \Phi(-1.78) - \Phi(-\infty) \] Since $ \Phi(-\infty) = 0 $, this simplifies to: \[ P(Z < -1.78) = \Phi(-1.78) \]

Step 3: Calculate the Probability

From the calculations, we find that: \[ \Phi(-1.78) = 0.0375 \] Thus, the area to the left of $ z = -1.78 $ under the standard normal curve is $ 0.0375 $.