High School

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Given a standardized normal distribution (with a mean of 0 and a standard deviation of 1), what is the probability that:

a. \( Z < 1.57 \)

b. \( Z > 1.84 \)

c. \( 1.57 < Z < 1.84 \)

e. \( Z = 1.57 \)

Answer :

a. P(Z < 1.57) is approximately 0.9418.

b. P(Z > 1.84) is approximately 0.0336.

c. P(1.57 < Z < 1.84) is approximately 0.0254.

d. P(Z = 1.57) is approximately zero.

To find the probabilities associated with the standardized normal distribution, we can use a standard normal distribution table or a statistical calculator. Here are the probabilities for the given scenarios:

  • a. P(Z < 1.57):

Using a standard normal distribution table or calculator, we find that the probability is approximately 0.9418.

  • b. P(Z > 1.84):

Similarly, using a standard normal distribution table or calculator, we find that the probability is approximately 0.0336.

  • c. P(1.57 < Z < 1.84):

To find this probability, we need to subtract the probability of Z < 1.57 from the probability of Z < 1.84. Using a standard normal distribution table or calculator, we have:

P(1.57 < Z < 1.84) = P(Z < 1.84) - P(Z < 1.57)

≈ 0.9672 - 0.9418

≈ 0.0254

  • d. P(Z = 1.57):

Since the normal distribution is a continuous probability distribution, the probability of Z taking on a specific value is zero.

Please note that the probabilities provided are approximations based on the standard normal distribution.

Learn more about probabilities

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