Answered

O Macmillan Learning
proportion:
The mean and standard deviation are set to 0 and 5,
respectively.
What proportion of the Normal density curve with this
mean and standard deviation falls below the value -10.0?
Enter your answer accurate to within three decimal places.
30
30

Verified Answer :

GinnyAnswer

Certainly! Let's solve the problem step-by-step:

1. Given Information:
- Mean (μ) = 0
- Standard Deviation (σ) = 5
- Value, \(\( x \)\) = -10.0

2. Calculate the Z-Score:
The Z-score formula is given by:
\(\[ Z = \frac{x - \mu}{\sigma} \]\)
Substitute the given values into the formula:
\(\[ Z = \frac{-10.0 - 0}{5} = \frac{-10.0}{5} = -2.0 \]\)
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 \(\( Z = -2.0 \)\), 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:
\(\[ 0.023 \]\)

So, the final answer is approximately 0.023 when rounded to three decimal places.

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