Solved on Mar 07, 2024

Find the probability of a standard normal random variable $z$ being between -1.65 and 0. Round the answer to four decimal places.

STEP 1

Assumptions
1. We are dealing with a standard normal distribution, which has a mean of 0 and a standard deviation of 1.
2. The variable $z$ represents a z-score, which is the number of standard deviations away from the mean.
3. $P(-1.65<z<0)$ represents the probability that the z-score is between -1.65 and 0.
4. We will use the standard normal distribution table or a calculator with statistical functions to find the probabilities associated with the given z-scores.

STEP 2

The probability $P(-1.65<z<0)$ can be found by calculating the area under the standard normal curve between the z-scores of -1.65 and 0.

STEP 3

First, we find the probability that $z$ is less than 0, which is the same as the probability that $z$ is less than the mean of the standard normal distribution.
$P(z<0)$

STEP 4

Since the standard normal distribution is symmetric about the mean, the probability that $z$ is less than 0 is equal to 0.5.
$P(z<0) = 0.5$

STEP 5

Next, we find the probability that $z$ is less than -1.65.
$P(z<-1.65)$

STEP 6

We look up the value of $P(z<-1.65)$ in the standard normal distribution table or use a calculator with statistical functions to find this probability.

STEP 7

The value from the standard normal distribution table for $z = -1.65$ is approximately 0.0495.
$P(z<-1.65) \approx 0.0495$

STEP 8

Now, we can find the probability that $z$ is between -1.65 and 0 by subtracting the probability that $z$ is less than -1.65 from the probability that $z$ is less than 0.
$P(-1.65<z<0) = P(z<0) - P(z<-1.65)$

STEP 9

Substitute the values we found into the equation.
$P(-1.65<z<0) = 0.5 - 0.0495$

STEP 10

Calculate the probability.
$P(-1.65<z<0) = 0.5 - 0.0495 = 0.4505$
The specified probability is approximately 0.4505. If necessary, round your answer to four decimal places.
$P(-1.65<z<0) \approx 0.4505$

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Find the probability of a standard normal random variable $z$ being between -1.65 and 0. Round the answer to four decimal places.

STEP 1

STEP 2

The probability $P(-1.65<z<0)$ can be found by calculating the area under the standard normal curve between the z-scores of -1.65 and 0.

STEP 3

First, we find the probability that $z$ is less than 0, which is the same as the probability that $z$ is less than the mean of the standard normal distribution.
$P(z<0)$

STEP 4

Since the standard normal distribution is symmetric about the mean, the probability that $z$ is less than 0 is equal to 0.5.
$P(z<0) = 0.5$

STEP 5

Next, we find the probability that $z$ is less than -1.65.
$P(z<-1.65)$

STEP 6

We look up the value of $P(z<-1.65)$ in the standard normal distribution table or use a calculator with statistical functions to find this probability.

STEP 7

The value from the standard normal distribution table for $z = -1.65$ is approximately 0.0495.
$P(z<-1.65) \approx 0.0495$

STEP 8

Now, we can find the probability that $z$ is between -1.65 and 0 by subtracting the probability that $z$ is less than -1.65 from the probability that $z$ is less than 0.
$P(-1.65<z<0) = P(z<0) - P(z<-1.65)$

STEP 9

Substitute the values we found into the equation.
$P(-1.65<z<0) = 0.5 - 0.0495$

STEP 10

Calculate the probability.
$P(-1.65<z<0) = 0.5 - 0.0495 = 0.4505$
The specified probability is approximately 0.4505. If necessary, round your answer to four decimal places.
$P(-1.65<z<0) \approx 0.4505$

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Find the probability of a standard normal random variable zzz being between -1.65 and 0. Round the answer to four decimal places.

STEP 1

STEP 2

The probability P(−1.65<z<0)P(-1.65<z<0)P(−1.65<z<0) can be found by calculating the area under the standard normal curve between the z-scores of -1.65 and 0.

STEP 3

First, we find the probability that zzz is less than 0, which is the same as the probability that zzz is less than the mean of the standard normal distribution.P(z<0)P(z<0)P(z<0)

STEP 4

Since the standard normal distribution is symmetric about the mean, the probability that zzz is less than 0 is equal to 0.5.P(z<0)=0.5P(z<0) = 0.5P(z<0)=0.5

STEP 5

Next, we find the probability that zzz is less than -1.65.P(z<−1.65)P(z<-1.65)P(z<−1.65)

STEP 6

We look up the value of P(z<−1.65)P(z<-1.65)P(z<−1.65) in the standard normal distribution table or use a calculator with statistical functions to find this probability.

STEP 7

The value from the standard normal distribution table for z=−1.65z = -1.65z=−1.65 is approximately 0.0495.P(z<−1.65)≈0.0495P(z<-1.65) \approx 0.0495P(z<−1.65)≈0.0495

STEP 8

Now, we can find the probability that zzz is between -1.65 and 0 by subtracting the probability that zzz is less than -1.65 from the probability that zzz is less than 0.P(−1.65<z<0)=P(z<0)−P(z<−1.65)P(-1.65<z<0) = P(z<0) - P(z<-1.65)P(−1.65<z<0)=P(z<0)−P(z<−1.65)

STEP 9

Substitute the values we found into the equation.P(−1.65<z<0)=0.5−0.0495P(-1.65<z<0) = 0.5 - 0.0495P(−1.65<z<0)=0.5−0.0495

STEP 10

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Find the probability of a standard normal random variable $z$ being between -1.65 and 0. Round the answer to four decimal places.

The probability $P(-1.65<z<0)$ can be found by calculating the area under the standard normal curve between the z-scores of -1.65 and 0.

First, we find the probability that $z$ is less than 0, which is the same as the probability that $z$ is less than the mean of the standard normal distribution.
$P(z<0)$

Since the standard normal distribution is symmetric about the mean, the probability that $z$ is less than 0 is equal to 0.5.
$P(z<0) = 0.5$

Next, we find the probability that $z$ is less than -1.65.
$P(z<-1.65)$

We look up the value of $P(z<-1.65)$ in the standard normal distribution table or use a calculator with statistical functions to find this probability.

The value from the standard normal distribution table for $z = -1.65$ is approximately 0.0495.
$P(z<-1.65) \approx 0.0495$

Now, we can find the probability that $z$ is between -1.65 and 0 by subtracting the probability that $z$ is less than -1.65 from the probability that $z$ is less than 0.
$P(-1.65<z<0) = P(z<0) - P(z<-1.65)$

Substitute the values we found into the equation.
$P(-1.65<z<0) = 0.5 - 0.0495$