Question

Given a standardized normal distribution​ (with a mean of 0 and a standard deviation of​ 1), determine the following probabilities.
a. ​P(Zgreater than1.07​)          b. ​P(Zless thannegative 0.25​)          c. ​P(minus1.96less thanZless thannegative 0.25​)
d. What is the value of Z if only 15.87​% of all possible​ Z-values are​ larger?

Answer 1

To determine the probabilities associated with a standardized normal distribution (Z-distribution), we'll use standard normal distribution tables or a calculator that can provide cumulative distribution function (CDF) values.

### a. \( P(Z > 1.07) \)

First, we need to find \( P(Z < 1.07) \) and then subtract it from 1:

Using a standard normal distribution table, or a calculator:
- \( P(Z < 1.07) \) is approximately \( 0.8577 \).

Thus,
\[
P(Z > 1.07) = 1 - P(Z < 1.07) = 1 - 0.8577 \approx 0.1423
\]

### b. \( P(Z < -0.25) \)

Using the standard normal distribution table:
- \( P(Z < -0.25) \) is approximately \( 0.4013 \).

### c. \( P(-1.96 < Z < -0.25) \)

To find this probability, we can calculate:
\[
P(-1.96 < Z < -0.25) = P(Z < -0.25) - P(Z < -1.96)
\]

From the standard normal distribution table:
- \( P(Z < -0.25) \approx 0.4013 \)
- \( P(Z < -1.96) \approx 0.0250 \)

Now we can compute the difference:
\[
P(-1.96 < Z < -0.25) = 0.4013 - 0.0250 = 0.3763
\]

### d. What is the value of Z if only 15.87% of all possible Z-values are larger?

To find this Z-value, we need to find the Z such that:
\[
P(Z > z) = 0.1587
\]

This is equivalent to finding:
\[
P(Z < z) = 1 - 0.1587 = 0.8413
\]

Using the standard normal distribution table or a calculator, we can find that:
- \( Z \approx 1.00 \) corresponds to the cumulative probability of \( 0.8413 \).

### Summary of Answers:
a. \( P(Z > 1.07) \approx 0.1423 \)
b. \( P(Z < -0.25) \approx 0.4013 \)
c. \( P(-1.96 < Z < -0.25) \approx 0.3763 \)
d. \( Z \approx 1.00 \)