Question

1. according to the normal curve shown, match each ( x )-value range to the correct percentage of data that falls in the interval.

10 to 30
15 to 25
( mu ) to ( mu + 2sigma )
15 to 20

choices
68%
34%
95%
47.5%

Explanation:

Step1: Identify the mean ($\mu$) and standard deviation ($\sigma$)

From the normal curve, the peak is at $x = 20$, so $\mu = 20$. The distance from $\mu$ to the first mark (15 or 25) is $20 - 15 = 5$, so $\sigma = 5$.

Step2: Analyze "10 to 30"

10 is $\mu - 2\sigma$ ($20 - 2\times5 = 10$) and 30 is $\mu + 2\sigma$ ($20 + 2\times5 = 30$). The percentage of data between $\mu - 2\sigma$ and $\mu + 2\sigma$ in a normal distribution is 95%.

Step3: Analyze "15 to 25"

15 is $\mu - \sigma$ ($20 - 5 = 15$) and 25 is $\mu + \sigma$ ($20 + 5 = 25$). The percentage of data between $\mu - \sigma$ and $\mu + \sigma$ is 68%.

Step4: Analyze "$\mu$ to $\mu + 2\sigma$"

$\mu = 20$, $\mu + 2\sigma = 30$. The area from $\mu$ to $\mu + 2\sigma$: The total area from $\mu - 2\sigma$ to $\mu + 2\sigma$ is 95%, so from $\mu$ to $\mu + 2\sigma$ (half of that plus the middle? Wait, no: The area from $\mu - 2\sigma$ to $\mu$ is 47.5% (since 95%/2), and from $\mu$ to $\mu + 2\sigma$ is also 47.5%? Wait, no, wait: Wait, the area from $\mu - \sigma$ to $\mu$ is 34%, $\mu$ to $\mu + \sigma$ is 34% (total 68% for $\mu - \sigma$ to $\mu + \sigma$). Then from $\mu - 2\sigma$ to $\mu - \sigma$ is 13.5%, $\mu + \sigma$ to $\mu + 2\sigma$ is 13.5% (so total 95% for $\mu - 2\sigma$ to $\mu + 2\sigma$: 13.5 + 34 + 34 + 13.5 = 95). So from $\mu$ to $\mu + 2\sigma$: $\mu$ to $\mu + \sigma$ is 34%, $\mu + \sigma$ to $\mu + 2\sigma$ is 13.5%, so total 34 + 13.5 = 47.5%.

Step5: Analyze "15 to 20"

15 is $\mu - \sigma$, 20 is $\mu$. The area from $\mu - \sigma$ to $\mu$ is 34% (since the total from $\mu - \sigma$ to $\mu + \sigma$ is 68%, so half is 34%).

Answer:

10 to 30: 95%
15 to 25: 68%
$\mu$ to $\mu + 2\sigma$: 47.5%
15 to 20: 34%