Math  /  Data & Statistics

QuestionFind the indicated probability using the standard normal distribution. P(z<1.78 or z>1.78)\mathrm{P}(\mathrm{z}<-1.78 \text { or } z>1.78)
Click here to view page 1 of the standard normal table. Click here to view page 2 of the standard normal table. P(z<1.78P(z<-1.78 or z>1.78)=z>1.78)= \square (Round to four decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? What's the chance that a randomly chosen value from a standard normal distribution is either smaller than 1.78-1.78 *or* bigger than 1.781.78? Watch out! Don't forget that "or" in probability usually means adding, and remember to use the properties of the standard normal distribution!

STEP 2

1. Find the probability of z<1.78z < -1.78.
2. Find the probability of z>1.78z > 1.78.
3. Add the probabilities.

STEP 3

Let's **look up** 1.78-1.78 in the **standard normal table**!
This gives us the probability of a value being *less* than 1.78-1.78.

STEP 4

The table tells us that P(z<1.78)=0.0375P(z < -1.78) = \mathbf{0.0375}.
Awesome!

STEP 5

The standard normal distribution is **symmetric** around zero.
This means P(z>1.78)P(z > 1.78) is the same as P(z<1.78)P(z < -1.78).
Why? Because the curve is a mirror image on either side of zero!

STEP 6

So, P(z>1.78)=P(z<1.78)=0.0375P(z > 1.78) = P(z < -1.78) = \mathbf{0.0375}.
Symmetry is super helpful!

STEP 7

Since we want the probability of zz being *less* than 1.78-1.78 *or* *greater* than 1.781.78, we **add** the two probabilities we just found.
Think of it like this: we're looking for the combined area in both tails of the distribution.

STEP 8

P(z<1.78 or z>1.78)=P(z<1.78)+P(z>1.78)P(z < -1.78 \text{ or } z > 1.78) = P(z < -1.78) + P(z > 1.78) =0.0375+0.0375= 0.0375 + 0.0375 =0.075= \mathbf{0.075}

STEP 9

The probability P(z<1.78P(z < -1.78 or z>1.78)z > 1.78) is 0.075\mathbf{0.075}.

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