Suppose Z follows the standard normal distribution. Use the calculator provided, or this table, to determine the value of c so that the following is true.
P(-1.24 ≤ Z≤ c)=0.8746                                                           Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.
 


To solve the problem, we are given that \( Z \) follows the standard normal distribution, and we are asked to find the value of \( c \) such that:

\[
P(-1.24 \leq Z \leq c) = 0.8746
\]

### Step 1: Find \( P(Z \leq -1.24) \)

Using a standard normal distribution table or a calculator, we first find the probability for \( Z = -1.24 \). From the standard normal table:

\[
P(Z \leq -1.24) = 0.1075
\]

### Step 2: Set up the equation

We are told that the probability between \( Z = -1.24 \) and \( Z = c \) is 0.8746. This can be written as:

\[
P(-1.24 \leq Z \leq c) = P(Z \leq c) - P(Z \leq -1.24)
\]

Substitute the known probability for \( Z \leq -1.24 \):

\[
0.8746 = P(Z \leq c) - 0.1075
\]

### Step 3: Solve for \( P(Z \leq c) \)

Add 0.1075 to both sides:

\[
P(Z \leq c) = 0.8746 + 0.1075 = 0.9821
\]

### Step 4: Find \( c \)

Now, we look up the value of \( c \) such that \( P(Z \leq c) = 0.9821 \) using the standard normal distribution table or a calculator.

From the table, \( Z = 2.08 \) corresponds to \( P(Z \leq 2.08) = 0.9821 \).

Thus, \( c = 2.08 \).

### Final Answer:
The value of \( c \) is approximately **2.08**.

---

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How do you use the standard normal table to find probabilities?
2. What is the relationship between cumulative probability and the Z-score?
3. How does changing the limits of integration affect the cumulative probability in a normal distribution?
4. What is the significance of the value of \( c \) in this problem?
5. How do you calculate probabilities for non-standard normal distributions?

**Tip**: When working with the standard normal distribution, always confirm whether you are working with left-tailed or right-tailed probabilities to avoid errors.

Suppose Z follows the standard normal distribution. Use the calculator provided, or this table, to determine the value of c so that the following is true. P(-1.24 ≤ Z ≤ c) = 0.8746. Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.

Determine the value of c in the standard normal distribution for the probability P(-1.24 ≤ Z ≤ c) = 0.8746. This detailed solution uses cumulative probability and Z-score calculations to find that c = 2.08. Suitable for students learning probability at an advanced high school or undergraduate level.

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