Find the indicated probability assuming that *x* is a random variable with a normal distribution with the given mean and standard deviation. (Round your answer to four decimal places.)

*P*(*x* ≤ 77), 𝜇 = 70, 𝜎 = 7

To find the probability \( P(x \leq 77) \) when \( x \) follows a normal distribution with mean \( \mu = 70 \) and standard deviation \( \sigma = 7 \), we can use the Z-score formula:

\[
Z = \frac{x - \mu}{\sigma}
\]

### Step 1: Calculate the Z-score

\[
Z = \frac{77 - 70}{7} = \frac{7}{7} = 1
\]

### Step 2: Find the cumulative probability

Now that we have the Z-score, we can find the probability corresponding to \( Z = 1 \). Using standard normal distribution tables or a calculator, the cumulative probability for \( Z = 1 \) is approximately:

\[
P(Z \leq 1) \approx 0.8413
\]

### Conclusion:
\[
P(x \leq 77) \approx 0.8413
\]

Thus, the probability is approximately **0.8413**.

Would you like further details on this calculation?

### Related Questions:
1. What is the probability \( P(x \geq 77) \) for the same normal distribution?
2. How would the probability change if \( x = 65 \) instead of \( x = 77 \)?
3. What is the probability \( P(70 \leq x \leq 77) \)?
4. Can you find \( P(x = 77) \) for this continuous distribution?
5. How would the result change if the standard deviation was 10 instead of 7?

### Tip:
When using the normal distribution, the Z-score helps you standardize the data, making it easier to reference probabilities from the standard normal table.

Find the indicated probability assuming that x is a random variable with a normal distribution with the given mean and standard deviation. (Round your answer to four decimal places.) P(x ≤ 77), μ = 70, σ = 7

Learn how to find the probability P(x ≤ 77) for a normal distribution with mean μ = 70 and standard deviation σ = 7. The problem uses the Z-score formula and standard normal distribution to calculate the cumulative probability, suitable for high school students in Grades 10-12.

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