Solved on Dec 13, 2023

Find the probability that a standard normal random variable $z$ is less than 0.24 or greater than or equal to -0.42.

STEP 1

Assumptions
1. We are dealing with a standard normal distribution, which is symmetric about the mean, $z=0$ .
2. We want to find the probability that $z$ is either less than $0.24$ or greater than or equal to $-0.42$ .
3. The probability of any event in a standard normal distribution can be found using the standard normal distribution table or a calculator with statistical functions.

STEP 2

The probability that $z$ is less than $0.24$ can be found directly using the standard normal distribution table or statistical functions on a calculator.
$P(z < 0.24)$

STEP 3

To find $P(z < 0.24)$ , we look up the value of $0.24$ in the standard normal distribution table or use a statistical calculator.

STEP 4

The probability that $z$ is greater than or equal to $-0.42$ can be found by subtracting the probability that $z$ is less than $-0.42$ from $1$ , since the total area under the standard normal curve is $1$ .
$P(z \geq -0.42) = 1 - P(z < -0.42)$

STEP 5

To find $P(z < -0.42)$ , we look up the value of $-0.42$ in the standard normal distribution table or use a statistical calculator. Remember that due to the symmetry of the standard normal distribution, $P(z < -0.42) = P(z > 0.42)$ .

STEP 6

Now we can calculate $P(z \geq -0.42)$ using the value found in the previous step.
$P(z \geq -0.42) = 1 - P(z < -0.42)$

STEP 7

The probability that $z$ is either less than $0.24$ or greater than or equal to $-0.42$ is the sum of the probabilities found in steps 3 and 6.
$P(z < 0.24 \text{ or } z \geq -0.42) = P(z < 0.24) + P(z \geq -0.42)$

STEP 8

Add the two probabilities together to find the final answer.
$P(z < 0.24 \text{ or } z \geq -0.42) = P(z < 0.24) + (1 - P(z < -0.42))$

STEP 9

Assuming we have found the values from the standard normal distribution table or a statistical calculator:
Let's say $P(z < 0.24) = a$ and $P(z < -0.42) = b$ , then we can write:
$P(z < 0.24 \text{ or } z \geq -0.42) = a + (1 - b)$

STEP 10

Now, plug in the values for $a$ and $b$ to get the numerical answer.
$P(z < 0.24 \text{ or } z \geq -0.42) = a + (1 - b)$
For example, if $a = 0.595$ and $b = 0.337$ , then:
$P(z < 0.24 \text{ or } z \geq -0.42) = 0.595 + (1 - 0.337)$

STEP 11

Calculate the final probability.
$P(z < 0.24 \text{ or } z \geq -0.42) = 0.595 + 0.663$
$P(z < 0.24 \text{ or } z \geq -0.42) = 1.258$
However, since the total probability cannot exceed $1$ , we must realize that there is an overlap in the events $z < 0.24$ and $z \geq -0.42$ . This overlap is exactly the probability $P(-0.42 \leq z < 0.24)$ , which we have mistakenly counted twice.

STEP 12

To correct the double-counting, we subtract the overlap probability $P(-0.42 \leq z < 0.24)$ from the sum we calculated.
$P(-0.42 \leq z < 0.24) = P(z < 0.24) - P(z < -0.42)$
$P(-0.42 \leq z < 0.24) = a - b$
$P(-0.42 \leq z < 0.24) = 0.595 - 0.337$
$P(-0.42 \leq z < 0.24) = 0.258$

STEP 13

Now, subtract the overlap probability from the sum we calculated in step 11.
$P(z < 0.24 \text{ or } z \geq -0.42) = 1.258 - 0.258$
$P(z < 0.24 \text{ or } z \geq -0.42) = 1.000$

STEP 14

The final probability is therefore $1.000$ , which makes sense because the events $z < 0.24$ and $z \geq -0.42$ cover the entire range of possibilities for a standard normal distribution.
$P(z < 0.24 \text{ or } z \geq -0.42) = 1.000$

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Find the probability that a standard normal random variable $z$ is less than 0.24 or greater than or equal to -0.42.

STEP 1

STEP 2

The probability that $z$ is less than $0.24$ can be found directly using the standard normal distribution table or statistical functions on a calculator.
$P(z < 0.24)$

STEP 3

To find $P(z < 0.24)$ , we look up the value of $0.24$ in the standard normal distribution table or use a statistical calculator.

STEP 4

The probability that $z$ is greater than or equal to $-0.42$ can be found by subtracting the probability that $z$ is less than $-0.42$ from $1$ , since the total area under the standard normal curve is $1$ .
$P(z \geq -0.42) = 1 - P(z < -0.42)$

STEP 5

To find $P(z < -0.42)$ , we look up the value of $-0.42$ in the standard normal distribution table or use a statistical calculator. Remember that due to the symmetry of the standard normal distribution, $P(z < -0.42) = P(z > 0.42)$ .

STEP 6

Now we can calculate $P(z \geq -0.42)$ using the value found in the previous step.
$P(z \geq -0.42) = 1 - P(z < -0.42)$

STEP 7

The probability that $z$ is either less than $0.24$ or greater than or equal to $-0.42$ is the sum of the probabilities found in steps 3 and 6.
$P(z < 0.24 \text{ or } z \geq -0.42) = P(z < 0.24) + P(z \geq -0.42)$

STEP 8

Add the two probabilities together to find the final answer.
$P(z < 0.24 \text{ or } z \geq -0.42) = P(z < 0.24) + (1 - P(z < -0.42))$

STEP 9

Assuming we have found the values from the standard normal distribution table or a statistical calculator:
Let's say $P(z < 0.24) = a$ and $P(z < -0.42) = b$ , then we can write:
$P(z < 0.24 \text{ or } z \geq -0.42) = a + (1 - b)$

STEP 10

Now, plug in the values for $a$ and $b$ to get the numerical answer.
$P(z < 0.24 \text{ or } z \geq -0.42) = a + (1 - b)$
For example, if $a = 0.595$ and $b = 0.337$ , then:
$P(z < 0.24 \text{ or } z \geq -0.42) = 0.595 + (1 - 0.337)$

STEP 11

STEP 12

To correct the double-counting, we subtract the overlap probability $P(-0.42 \leq z < 0.24)$ from the sum we calculated.
$P(-0.42 \leq z < 0.24) = P(z < 0.24) - P(z < -0.42)$
$P(-0.42 \leq z < 0.24) = a - b$
$P(-0.42 \leq z < 0.24) = 0.595 - 0.337$
$P(-0.42 \leq z < 0.24) = 0.258$

STEP 13

Now, subtract the overlap probability from the sum we calculated in step 11.
$P(z < 0.24 \text{ or } z \geq -0.42) = 1.258 - 0.258$
$P(z < 0.24 \text{ or } z \geq -0.42) = 1.000$

STEP 14

The final probability is therefore $1.000$ , which makes sense because the events $z < 0.24$ and $z \geq -0.42$ cover the entire range of possibilities for a standard normal distribution.
$P(z < 0.24 \text{ or } z \geq -0.42) = 1.000$

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Find the probability that a standard normal random variable zzz is less than 0.24 or greater than or equal to -0.42.

STEP 1

STEP 2

The probability that zzz is less than 0.240.240.24 can be found directly using the standard normal distribution table or statistical functions on a calculator.P(z<0.24)P(z < 0.24)P(z<0.24)

STEP 3

To find P(z<0.24)P(z < 0.24)P(z<0.24), we look up the value of 0.240.240.24 in the standard normal distribution table or use a statistical calculator.

STEP 4

STEP 5

STEP 6

Now we can calculate P(z≥−0.42)P(z \geq -0.42)P(z≥−0.42) using the value found in the previous step.P(z≥−0.42)=1−P(z<−0.42)P(z \geq -0.42) = 1 - P(z < -0.42)P(z≥−0.42)=1−P(z<−0.42)

STEP 7

STEP 8

Add the two probabilities together to find the final answer.P(z<0.24 or z≥−0.42)=P(z<0.24)+(1−P(z<−0.42))P(z < 0.24 \text{ or } z \geq -0.42) = P(z < 0.24) + (1 - P(z < -0.42))P(z<0.24 or z≥−0.42)=P(z<0.24)+(1−P(z<−0.42))

STEP 9

STEP 10

STEP 11

STEP 12

STEP 13

STEP 14

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Find the probability that a standard normal random variable $z$ is less than 0.24 or greater than or equal to -0.42.

The probability that $z$ is less than $0.24$ can be found directly using the standard normal distribution table or statistical functions on a calculator.
$P(z < 0.24)$

To find $P(z < 0.24)$ , we look up the value of $0.24$ in the standard normal distribution table or use a statistical calculator.

Now we can calculate $P(z \geq -0.42)$ using the value found in the previous step.
$P(z \geq -0.42) = 1 - P(z < -0.42)$

Add the two probabilities together to find the final answer.
$P(z < 0.24 \text{ or } z \geq -0.42) = P(z < 0.24) + (1 - P(z < -0.42))$