Question

fill in the blanks with the percentage of the area under the normal curve that each section represents. $\bar{x}-3sigma$ $\bar{x}-2sigma$ $\bar{x}-sigma$ $\bar{x}$ $\bar{x}+sigma$ $\bar{x}+2sigma$ $\bar{x}+3sigma$ 68% 34% 17% 8.5% 47.5% 2.5% 13.5% 15% 0.3% 2.35% 0.15% 5% 32% 12.5%

Explanation:

Step1: Recall normal - curve properties

The total area under the normal curve is 100%. The 68 - 95 - 99.7 rule states that about 68% of the data lies within 1 standard deviation ($\mu\pm\sigma$), about 95% lies within 2 standard deviations ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard deviations ($\mu\pm3\sigma$).

Step2: Calculate areas between intervals

• The area between $\bar{x}-\sigma$ and $\bar{x}$ is the same as the area between $\bar{x}$ and $\bar{x}+\sigma$. Since the area between $\bar{x}-\sigma$ and $\bar{x}+\sigma$ is 68%, the area between $\bar{x}-\sigma$ and $\bar{x}$ (or $\bar{x}$ and $\bar{x}+\sigma$) is $\frac{68\%}{2}=34\%$.
• The area between $\bar{x}-2\sigma$ and $\bar{x}-\sigma$ is the same as the area between $\bar{x}+\sigma$ and $\bar{x}+2\sigma$. The area between $\bar{x}-2\sigma$ and $\bar{x}+2\sigma$ is 95%. So the area outside of $\bar{x}-2\sigma$ and $\bar{x}+2\sigma$ but within $\bar{x}-3\sigma$ and $\bar{x}+3\sigma$ is $99.7 - 95=4.7\%$. Then the area between $\bar{x}-2\sigma$ and $\bar{x}-\sigma$ (or $\bar{x}+\sigma$ and $\bar{x}+2\sigma$) is $\frac{95 - 68}{2}=13.5\%$.
• The area between $\bar{x}-3\sigma$ and $\bar{x}-2\sigma$ is the same as the area between $\bar{x}+2\sigma$ and $\bar{x}+3\sigma$. The area outside of $\bar{x}-3\sigma$ and $\bar{x}+3\sigma$ is $100 - 99.7 = 0.3\%$. So the area between $\bar{x}-3\sigma$ and $\bar{x}-2\sigma$ (or $\bar{x}+2\sigma$ and $\bar{x}+3\sigma$) is $\frac{100 - 99.7}{2}=0.15\%$.

From left - to - right:

• The area between $\bar{x}-3\sigma$ and $\bar{x}-2\sigma$ is 2.35%
• The area between $\bar{x}-2\sigma$ and $\bar{x}-\sigma$ is 13.5%
• The area between $\bar{x}-\sigma$ and $\bar{x}$ is 34%
• The area between $\bar{x}$ and $\bar{x}+\sigma$ is 34%
• The area between $\bar{x}+\sigma$ and $\bar{x}+2\sigma$ is 13.5%
• The area between $\bar{x}+2\sigma$ and $\bar{x}+3\sigma$ is 2.35%

Answer:

From left - to - right: 0.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, 0.15%