Question

Sam has 6 rose bushes. He counted the flowers on each of them. There are 8, 2, 5, 4, 11 and 9. Find the Standard Deviation. Is X a “Usual” number of flowers? X is your last two digits of your CUNYFirst ID Number. (00 = 0, 01 = 1, …) (20 points) my id is 03

Answer 1

$$mean=\bar{x}=\dfrac{8+2+5+4+11+9}{6}=6.5$$

$$Var(x)=\sigma^2=\dfrac{1}{6}((8-6.5)^2+(2-6.5)^2$$

$$+(5-6.5)^2+(4-6.5)^2+(11-6.5)^2+(9-6.5)^2)$$

$$=\dfrac{57.5}{6}\approx9.583333$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{57.5}{6}}\approx3.0957$$

$$\mu-\sigma=6.5-\sqrt{\dfrac{57.5}{6}}\approx3.4043$$

$$\mu+\sigma=6.5+\sqrt{\dfrac{57.5}{6}}\approx9.5957$$

By 68–95–99.7 rule

$$P(\mu-2\sigma\leq x\leq \mu+2\sigma)=0.9545$$

$$\mu-2\sigma=6.5-2(\sqrt{\dfrac{57.5}{6}})\approx0.3086$$

$$\mu+2\sigma=6.5+2(\sqrt{\dfrac{57.5}{6}})\approx12.6914$$

$3$ is within $[\mu-2\sigma, \mu+2\sigma].$ Then  $X=3$ is a “Usual” number of flowers.