Math / Data & Statistics

QuestionUse this table or the ALEKS calculator to complete the following. Give your answers to four decimal places (for example, 0.1234). (a) Find the area under the standard normal curve to the right of $z=-0.45$ . $0.3264$ (b) Find the area under the standard normal curve between $z=0.82$ and $z=2.20$ . $-0.1922$

Studdy Solution

STEP 1

What is this asking? We need to find areas under the standard normal curve, which basically means finding probabilities for a normal distribution with mean $0$ and standard deviation $1$ .
One area is to the right of a negative z-score, and the other is between two positive z-scores. Watch out! Make sure to use the correct z-scores and understand what "to the right" and "between" mean in terms of area.
Also, remember areas (probabilities) can't be negative!

STEP 2

1. Area to the Right
2. Area Between

STEP 3

We want the area to the right of $z = -0.45$ .
This means we're looking for $P(Z > -0.45)$ , where $Z$ is our standard normal random variable.

STEP 4

Because the standard normal distribution is symmetric around $0$ , the area to the right of $z = -0.45$ is the same as the area to the left of $z = +0.45$ .
Think of it like folding the curve in half at zero; the areas on either side of the fold line match up!
So, $P(Z > -0.45) = P(Z < 0.45)$ .

STEP 5

Now we can look up $P(Z < 0.45)$ in a z-table or use a calculator.
We find that $P(Z < 0.45) \approx 0.6736$ .

STEP 6

We're looking for the area under the curve between $z = 0.82$ and $z = 2.20$ .
That's $P(0.82 < Z < 2.20)$ .

STEP 7

To find the area between two z-scores, we can find the area to the left of the larger z-score and subtract the area to the left of the smaller z-score.
This is like finding the area of a bigger rectangle and cutting out the area of a smaller rectangle to leave just the part we want in the middle.
So, $P(0.82 < Z < 2.20) = P(Z < 2.20) - P(Z < 0.82)$ .

STEP 8

Looking up the values in a z-table or using a calculator, we find $P(Z < 2.20) \approx 0.9861$ and $P(Z < 0.82) \approx 0.7939$ .

STEP 9

Now, subtract: $0.9861 - 0.7939 = 0.1922$ .
So, $P(0.82 < Z < 2.20) \approx 0.1922$ .

STEP 10

(a) The area to the right of $z = -0.45$ is approximately 0.6736. (b) The area under the standard normal curve between $z = 0.82$ and $z = 2.20$ is approximately 0.1922.

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Studdy Solution

STEP 1

STEP 2

1. Area to the Right
2. Area Between

STEP 3

We want the area to the right of $z = -0.45$ .
This means we're looking for $P(Z > -0.45)$ , where $Z$ is our standard normal random variable.

STEP 4

Because the standard normal distribution is symmetric around $0$ , the area to the right of $z = -0.45$ is the same as the area to the left of $z = +0.45$ .
Think of it like folding the curve in half at zero; the areas on either side of the fold line match up!
So, $P(Z > -0.45) = P(Z < 0.45)$ .

STEP 5

Now we can look up $P(Z < 0.45)$ in a z-table or use a calculator.
We find that $P(Z < 0.45) \approx 0.6736$ .

STEP 6

We're looking for the area under the curve between $z = 0.82$ and $z = 2.20$ .
That's $P(0.82 < Z < 2.20)$ .

STEP 7

STEP 8

Looking up the values in a z-table or using a calculator, we find $P(Z < 2.20) \approx 0.9861$ and $P(Z < 0.82) \approx 0.7939$ .

STEP 9

Now, subtract: $0.9861 - 0.7939 = 0.1922$ .
So, $P(0.82 < Z < 2.20) \approx 0.1922$ .

STEP 10

(a) The area to the right of $z = -0.45$ is approximately 0.6736. (b) The area under the standard normal curve between $z = 0.82$ and $z = 2.20$ is approximately 0.1922.

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Studdy Solution

STEP 1

STEP 2

1. Area to the Right2. Area Between

STEP 3

We want the area to the right of z=−0.45z = -0.45z=−0.45.This means we're looking for P(Z>−0.45)P(Z > -0.45)P(Z>−0.45), where ZZZ is our standard normal random variable.

STEP 4

STEP 5

Now we can look up P(Z<0.45)P(Z < 0.45)P(Z<0.45) in a z-table or use a calculator.We find that P(Z<0.45)≈0.6736P(Z < 0.45) \approx 0.6736P(Z<0.45)≈0.6736.

STEP 6

We're looking for the area under the curve between z=0.82z = 0.82z=0.82 and z=2.20z = 2.20z=2.20.That's P(0.82<Z<2.20)P(0.82 < Z < 2.20)P(0.82<Z<2.20).

STEP 7

STEP 8

Looking up the values in a z-table or using a calculator, we find P(Z<2.20)≈0.9861P(Z < 2.20) \approx 0.9861P(Z<2.20)≈0.9861 and P(Z<0.82)≈0.7939P(Z < 0.82) \approx 0.7939P(Z<0.82)≈0.7939.

STEP 9

Now, subtract: 0.9861−0.7939=0.19220.9861 - 0.7939 = 0.19220.9861−0.7939=0.1922.So, P(0.82<Z<2.20)≈0.1922P(0.82 < Z < 2.20) \approx 0.1922P(0.82<Z<2.20)≈0.1922.

STEP 10

(a) The area to the right of z=−0.45z = -0.45z=−0.45 is approximately **0.6736**. (b) The area under the standard normal curve between z=0.82z = 0.82z=0.82 and z=2.20z = 2.20z=2.20 is approximately **0.1922**.

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1. Area to the Right
2. Area Between

We want the area to the right of $z = -0.45$ .
This means we're looking for $P(Z > -0.45)$ , where $Z$ is our standard normal random variable.

Now we can look up $P(Z < 0.45)$ in a z-table or use a calculator.
We find that $P(Z < 0.45) \approx 0.6736$ .

We're looking for the area under the curve between $z = 0.82$ and $z = 2.20$ .
That's $P(0.82 < Z < 2.20)$ .

Looking up the values in a z-table or using a calculator, we find $P(Z < 2.20) \approx 0.9861$ and $P(Z < 0.82) \approx 0.7939$ .

Now, subtract: $0.9861 - 0.7939 = 0.1922$ .
So, $P(0.82 < Z < 2.20) \approx 0.1922$ .

(a) The area to the right of $z = -0.45$ is approximately 0.6736. (b) The area under the standard normal curve between $z = 0.82$ and $z = 2.20$ is approximately 0.1922.