Question

josiah counted the number of silver beads on each bracelet at hampton jewelry, the store where he works.

number of silver beads	number of bracelets
44	3
45	2
54	4
82	6
89	2
91	2

x is the number of silver beads that a randomly chosen bracelet had. what is the variance of x?
write your answer as a decimal.

Explanation:

Step1: Calculate the total number of bracelets

$n=1 + 3+2 + 4+6+2+2=20$

Step2: Calculate the expected - value (mean) $\mu$

$\mu=\frac{37\times1 + 44\times3+45\times2 + 54\times4+82\times6+89\times2+91\times2}{20}$
$=\frac{37+132 + 90+216+492+178+182}{20}$
$=\frac{1327}{20}=66.35$

Step3: Calculate the variance $\sigma^{2}$

$\sigma^{2}=\frac{1\times(37 - 66.35)^{2}+3\times(44 - 66.35)^{2}+2\times(45 - 66.35)^{2}+4\times(54 - 66.35)^{2}+6\times(82 - 66.35)^{2}+2\times(89 - 66.35)^{2}+2\times(91 - 66.35)^{2}}{20}$
$=\frac{1\times(-29.35)^{2}+3\times(-22.35)^{2}+2\times(-21.35)^{2}+4\times(-12.35)^{2}+6\times(15.65)^{2}+2\times(22.65)^{2}+2\times(24.65)^{2}}{20}$
$=\frac{1\times861.4225+3\times499.5225+2\times455.8225+4\times152.5225+6\times244.9225+2\times512.0225+2\times607.6225}{20}$
$=\frac{861.4225 + 1498.5675+911.645+610.09+1469.535+1024.045+1215.245}{20}$
$=\frac{7590.55}{20}=379.5275$

Answer:

$379.5275$