Answer :
a. P(Z < 1.57) ≈ 0.9429 - The probability that Z is less than 0.57.
b. P(Z > 1.88) ≈ 0.0314 - The probability that Z is greater than 0.88.
c. P(1.57 < Z < 1.88) ≈ 0.0257 - The probability that Z is between 1.57 and 0.88.
d. P(Z < 1.57 or Z > 1.88) ≈ 0.9743 - The probability that Z is either less than 1.57 or greater than 0.88.
To find these probabilities for a standard normal distribution, you can use a standard normal distribution table or a calculator. Here are the calculations for each part:
a. Probability that Z is less than 1.57:
You want to find P(Z < 1.57). Using a standard normal distribution table or calculator, you can find that P(Z < 1.57) is approximately 0.9429 (rounded to four decimal places).
b. Probability that Z is greater than 1.88:
You want to find P(Z > 1.88). Since the standard normal distribution is symmetric, P(Z > 1.88) is the same as 1 - P(Z < 1.88). Using a standard normal distribution table or calculator, you can find that P(Z < 1.88) is approximately 0.9686. So, P(Z > 1.88) is 1 - 0.9686 = 0.0314 (rounded to four decimal places).
c. Probability that Z is between 1.57 and 1.88:
You want to find P(1.57 < Z < 1.88). You can do this by subtracting P(Z < 1.57) from P(Z < 1.88):
P(1.57 < Z < 1.88) = P(Z < 1.88) - P(Z < 1.57)
P(1.57 < Z < 1.88) ≈ 0.9686 - 0.9429 ≈ 0.0257 (rounded to four decimal places).
d. Probability that Z is less than 1.57 or greater than 1.88:
You want to find P(Z < 1.57 or Z > 1.88). This can be calculated by adding P(Z < 1.57) to P(Z > 1.88):
P(Z < 1.57 or Z > 1.88) = P(Z < 1.57) + P(Z > 1.88) ≈ 0.9429 + 0.0314 ≈ 0.9743 (rounded to four decimal places).
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