Let $$ Z \sim N(0,1) $$. Accurate to 4 decimal places, evaluate $$ P(-2.58

Question

UpStudy AI · Accepted Answer

We are given that $$ Z \sim N(0,1) $$ and we want to find

$$
P(-2.58 < Z < 2.65).
$$

**Step 1. Express the probability in terms of the cumulative distribution function (CDF) $$\Phi(z)$$:**

$$
P(-2.58 < Z < 2.65) = \Phi(2.65) - \Phi(-2.58).
$$

**Step 2. Use the symmetry of the normal distribution:**

Since

$$
\Phi(-a) = 1 - \Phi(a),
$$

we have

$$
\Phi(-2.58) = 1 - \Phi(2.58).
$$

Thus, the probability becomes:

$$
P(-2.58 < Z < 2.65) = \Phi(2.65) + \Phi(2.58) - 1.
$$

**Step 3. Find the values of $$\Phi(2.65)$$ and $$\Phi(2.58)$$:**

Using standard normal tables or a calculator:
- $$\Phi(2.65) \approx 0.9960$$
- $$\Phi(2.58) \approx 0.9951$$

**Step 4. Compute the probability:**

$$
P(-2.58 < Z < 2.65) \approx 0.9960 + 0.9951 - 1 = 1.9911 - 1 = 0.9911.
$$

Rounded to four decimal places, we have:

$$
P(-2.58 < Z < 2.65) \approx 0.9911.
$$

Thus, the correct answer is:

$$
\boxed{0.991}
$$
The probability $$ P(-2.58 < Z < 2.65) $$ is approximately 0.991.

Let \( Z \sim N(0,1) \). Accurate to 4 decimal places, evaluate \( P(-2.58

Ask by Estrada Coles. in the United States

Mar 09,2025