Here are the step-by-step solutions for each problem based on the properties of the Normal Distribution curve.
Problem 4
Goal: Find the percent of the area in the shaded region between $\mu - 3\sigma$ and $\mu - \sigma$.
1.
Identify the sections: The shaded area covers two specific "slices" of the bell curve to the left of the center:
* The slice between $-3\sigma$ and $-2\sigma$.
* The slice between $-2\sigma$ and $-1\sigma$.
2.
Recall the percentages: In a standard normal distribution:
* The area between $-3\sigma$ and $-2\sigma$ is approximately
2.35%.
* The area between $-2\sigma$ and $-1\sigma$ is approximately
13.5%.
3.
Add them together:
$$2.35\% + 13.5\% = 15.85\%$$
Answer for #4: 15.85%
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Problem 6
Goal: Find the total percent of the area in the two separate shaded regions.
1.
Identify the left shaded region: This is the area between $\mu - 2\sigma$ and $\mu - \sigma$.
* From our standard values, this area is
13.5%.
2.
Identify the right shaded region: This is the area to the right of $\mu + \sigma$ (everything past the first mark on the right).
* We know the area from the center ($\mu$) to $+1\sigma$ is 34%.
* The entire right half of the curve is 50%.
* So, the tail end is $50\% - 34\% = \mathbf{16\%}$.
3.
Add them together:
$$13.5\% + 16\% = 29.5\%$$
Answer for #6: 29.5%
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Problem 10
Goal: Find $P(x \leq \mu + \sigma)$. This asks for the total area under the curve to the left of the point $\mu + \sigma$.
1.
Break it down: The area to the left of $\mu + \sigma$ consists of two parts:
* The entire left half of the curve (from far left up to the mean $\mu$). This is always
50%.
* The section from the mean $\mu$ to $+1\sigma$. This is
34%.
2.
Add them together:
$$50\% + 34\% = 84\%$$
Answer for #10: 84% (or 0.84)
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Problem 12
Goal: Find $P(\mu - 3\sigma \leq x \leq \mu)$. This asks for the area between $-3\sigma$ and the center mean $\mu$.
1.
Method 1 (Subtraction):
* The entire left half of the curve (from $-\infty$ to $\mu$) is
50%.
* The tiny unshaded tail to the left of $-3\sigma$ is
0.15%.
* Subtract the tail from the half: $50\% - 0.15\% = 49.85\%$.
2.
Method 2 (Addition):
* Add the three slices from left to center:
* $-3\sigma$ to $-2\sigma$:
2.35%
* $-2\sigma$ to $-1\sigma$:
13.5%
* $-1\sigma$ to $\mu$:
34%
* Sum: $2.35 + 13.5 + 34 = 49.85\%$.
Answer for #12: 49.85% (or 0.4985)
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Final Answer:
4. 15.85%
6. 29.5%
10. 84%
12. 49.85%
Parent Tip: Review the logic above to help your child master the concept of algebra 2 normal distribution worksheet.
