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.