Given that z is a standard normal random variable, compute the following probabilities (to 4 decimals).
1. P(-1.92 ≤ z ≤ 0.43)
2.P(0.57 ≤ z ≤ 1.24)
3.P(-1.75 ≤ z ≤ -1.02)
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To compute the probabilities, we will use the standard normal distribution table or a calculator that can provide values for the standard normal distribution.
1. P(-1.92 ≤ z ≤ 0.43):
Using the standard normal distribution table or calculator, we find:
P(z ≤ 0.43) = 0.6664
P(z ≤ -1.92) = 0.0274
Therefore,
P(-1.92 ≤ z ≤ 0.43) = P(z ≤ 0.43) - P(z ≤ -1.92)
= 0.6664 - 0.0274
= 0.6390
So, the probability P(-1.92 ≤ z ≤ 0.43) is approximately 0.6390.
2. P(0.57 ≤ z ≤ 1.24):
Using the standard normal distribution table or calculator, we find:
P(z ≤ 0.57) = 0.7131
P(z ≤ 1.24) = 0.8925
Therefore,
P(0.57 ≤ z ≤ 1.24) = P(z ≤ 1.24) - P(z ≤ 0.57)
= 0.8925 - 0.7131
= 0.1794
So, the probability P(0.57 ≤ z ≤ 1.24) is approximately 0.1794.
3. P(-1.75 ≤ z ≤ -1.02):
Using the standard normal distribution table or calculator, we find:
P(z ≤ -1.75) = 0.0401
P(z ≤ -1.02) = 0.1549
Therefore,
P(-1.75 ≤ z ≤ -1.02) = P(z ≤ -1.02) - P(z ≤ -1.75)
= 0.1549 - 0.0401
= 0.1148
So, the probability P(-1.75 ≤ z ≤ -1.02) is approximately 0.1148.
To compute these probabilities, we can use the standard normal distribution table or a statistical software or calculator.
1. P(-1.92 ≤ z ≤ 0.43)
First, find the z-score for -1.92 and 0.43. Using a standard normal distribution table or a calculator, we can find that the z-score for -1.92 is approximately -0.9732 and the z-score for 0.43 is approximately 0.6664.
Then, we calculate the probability by subtracting the value of the cumulative distribution function (CDF) for the lower z-score from the CDF for the higher z-score.
P(-1.92 ≤ z ≤ 0.43) = P(z ≤ 0.43) - P(z ≤ -1.92)
= 0.6664 - 0.1645
≈ 0.5019
Therefore, P(-1.92 ≤ z ≤ 0.43) is approximately 0.5019.
2. P(0.57 ≤ z ≤ 1.24)
Similarly, find the z-score for 0.57 and 1.24. Using a standard normal distribution table or a calculator, the z-score for 0.57 is approximately 0.1431 and the z-score for 1.24 is approximately 0.8925.
Then, calculate the probability by subtracting the CDF for the lower z-score from the CDF for the higher z-score.
P(0.57 ≤ z ≤ 1.24) = P(z ≤ 1.24) - P(z ≤ 0.57)
= 0.8925 - 0.5636
≈ 0.3289
Therefore, P(0.57 ≤ z ≤ 1.24) is approximately 0.3289.
3. P(-1.75 ≤ z ≤ -1.02)
Find the z-score for -1.75 and -1.02. Using a standard normal distribution table or a calculator, the z-score for -1.75 is approximately -0.9664, and the z-score for -1.02 is approximately -0.3365.
Then, calculate the probability by subtracting the CDF for the lower z-score from the CDF for the higher z-score.
P(-1.75 ≤ z ≤ -1.02) = P(z ≤ -1.02) - P(z ≤ -1.75)
= 0.3413 - 0.1990
≈ 0.1423
Therefore, P(-1.75 ≤ z ≤ -1.02) is approximately 0.1423.