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

josiah counted the number of silver beads on each bracelet at hampton jewelry, the store where he works.\n|number of silver beads|number of bracelets|\n|----|----|\n|37|1|\n|44|3|\n|45|2|\n|54|4|\n|82|6|\n|89|2|\n|91|2|\nx is the number of silver beads that a randomly chosen bracelet had. what is the variance of x?\nwrite your answer as a decimal.

josiah counted the number of silver beads on each bracelet at hampton jewelry, the store where he works.\n|number of silver beads|number of bracelets|\n|----|----|\n|37|1|\n|44|3|\n|45|2|\n|54|4|\n|82|6|\n|89|2|\n|91|2|\nx is the number of silver beads that a randomly chosen bracelet had. what is the variance of x?\nwrite your answer as a decimal.

Answer

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:

$379.5275$