$$ X $$ is a normally distributed random variable with mean 34 and standard deviation 3 . What is the probability that $$ X $$ is less than 43 ? Use the $$ 0.68-0.95-0.997 $$ rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

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To determine the probability that $$ X $$ is less than 43, given that $$ X $$ is normally distributed with a mean ($$ \mu $$) of 34 and a standard deviation ($$ \sigma $$) of 3, we can use the 68-95-99.7 (empirical) rule.

### Steps:

1. **Identify the Number of Standard Deviations:**
   $$
   43 = \mu + k\sigma \implies 43 = 34 + k \times 3 \implies k = \frac{43 - 34}{3} = 3
   $$
   So, 43 is 3 standard deviations above the mean.

2. **Apply the Empirical Rule:**
   - **Within 1 standard deviation ($$ \mu \pm 1\sigma $$):** Approximately 68% of the data.
   - **Within 2 standard deviations ($$ \mu \pm 2\sigma $$):** Approximately 95% of the data.
   - **Within 3 standard deviations ($$ \mu \pm 3\sigma $$):** Approximately 99.7% of the data.

3. **Interpret the Result:**
   Since 43 is $$ \mu + 3\sigma $$, it represents the upper bound of the 99.7% range. This means that approximately 99.7% of the data lies below 43.

### Final Answer:
$$
0.997
$$
The probability that $$ X $$ is less than 43 is 0.997.

\( X \) is a normally distributed random variable with mean 34 and standard deviation 3 . What is the probability that \( X \) is less than 43 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

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