Question

1. scores on the math section of the sat are normally distributed with a mean of 500 and a standard deviation of 100. what percentage of sat math scores are between: (5 points)

400 and 700\ta. 34%
400 and 600\tb. 95%
500 and 700\tc. 81.5%
300 and 700\td. 68%
400 and 500\te. 47.5%

Explanation:

Step1: Recall the properties of normal distribution

In a normal - distribution, about 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$), about 95% lies within 2 standard deviations of the mean ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard deviations of the mean ($\mu\pm3\sigma$). Given $\mu = 500$ and $\sigma=100$.

Step2: Calculate the z - scores and find the percentages for each interval

Interval 400 and 700:

The z - score is calculated as $z=\frac{x - \mu}{\sigma}$. For $x = 400$, $z_1=\frac{400 - 500}{100}=- 1$; for $x = 700$, $z_2=\frac{700 - 500}{100}=2$. The percentage of data between $z=-1$ and $z = 2$ is $34\%+47.5\%=81.5\%$.

Interval 400 and 600:

For $x = 400$, $z_1=-1$; for $x = 600$, $z_2=\frac{600 - 500}{100}=1$. The percentage of data between $z=-1$ and $z = 1$ is $34\%+34\% = 68\%$.

Interval 500 and 700:

For $x = 500$, $z_1 = 0$; for $x = 700$, $z_2=2$. The percentage of data between $z = 0$ and $z = 2$ is $47.5\%$.

Interval 300 and 700:

For $x = 300$, $z_1=\frac{300 - 500}{100}=-2$; for $x = 700$, $z_2 = 2$. The percentage of data between $z=-2$ and $z = 2$ is $95\%$.

Interval 400 and 500:

For $x = 400$, $z_1=-1$; for $x = 500$, $z_2 = 0$. The percentage of data between $z=-1$ and $z = 0$ is $34\%$.

Answer:

400 and 700: c. 81.5%
400 and 600: d. 68%
500 and 700: e. 47.5%
300 and 700: b. 95%
400 and 500: a. 34%