Certainly! Let's solve the problem step-by-step:
1. Given Information:
- Mean (μ) = 0
- Standard Deviation (σ) = 5
- Value, [tex]\( x \)[/tex] = -10.0
2. Calculate the Z-Score:
The Z-score formula is given by:
[tex]\[
Z = \frac{x - \mu}{\sigma}
\][/tex]
Substitute the given values into the formula:
[tex]\[
Z = \frac{-10.0 - 0}{5} = \frac{-10.0}{5} = -2.0
\][/tex]
So, the Z-score for the value -10.0 is -2.0.
3. Find the Proportion (Cumulative Probability):
Using the Z-score obtained, we need to find the cumulative probability or the proportion of the distribution that falls below this Z-score.
The cumulative distribution function (CDF) for the standard normal distribution gives us this cumulative probability. For [tex]\( Z = -2.0 \)[/tex], we find that the proportion below this value is 0.023.
4. Conclusion:
The proportion of the Normal density curve with a mean of 0 and a standard deviation of 5 that falls below the value -10.0 is 0.023.
Hence, the proportion is:
[tex]\[
0.023
\][/tex]
So, the final answer is approximately 0.023 when rounded to three decimal places.
