Solved on Mar 07, 2024

Find the probability of a standard normal random variable zz 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 zz represents a z-score, which is the number of standard deviations away from the mean.
3. P(1.65<z<0)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)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 zz is less than 0, which is the same as the probability that zz is less than the mean of the standard normal distribution.
P(z<0)P(z<0)

STEP 4

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

STEP 5

Next, we find the probability that zz is less than -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) 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 = -1.65 is approximately 0.0495.
P(z<1.65)0.0495P(z<-1.65) \approx 0.0495

STEP 8

Now, we can find the probability that zz is between -1.65 and 0 by subtracting the probability that zz is less than -1.65 from the probability that zz 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)

STEP 9

Substitute the values we found into the equation.
P(1.65<z<0)=0.50.0495P(-1.65<z<0) = 0.5 - 0.0495

STEP 10

Calculate the probability.
P(1.65<z<0)=0.50.0495=0.4505P(-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)0.4505P(-1.65<z<0) \approx 0.4505

Was this helpful?
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ContactInfluencer programPolicyTerms
TwitterInstagramFacebookTikTokDiscord