Question

The weights of cockroaches living in a typical college dormitory are approximately distributed with a mean of 80 grams and a standard deviation of 4 grams. The percentage of cockroaches weighing between 77 grams and 83 grams is about _______%. Round to the nearest whole number.

Answer 1

Answer:

The percentage of cockroaches weighing between 77 grams and 83 grams is about 55%.

Step-by-step explanation:

Given : The weights of cockroaches living in a typical college dormitory are approximately distributed with a mean of 80 grams and a standard deviation of 4 grams.

To find : The percentage of cockroaches weighing between 77 grams and 83 grams is about __% ?

Solution :

Applying z-score formula,

$z=\frac{x-\mu}{\sigma}$

Where, x is the sample mean

$\mu=80$ is the population mean

$\sigma=4$ is the standard deviation.

For x=77 z-score is

$z=\frac{77-80}{4}$

$z=\frac{-3}{4}$

$z=-0.75$

For x=83 z-score is

$z=\frac{83-80}{4}$

$z=\frac{3}{4}$

$z=0.75$

The probability of z lie between 77 and 83 is given by z-score table,

$P(z<77)=0.2266$

$P(z<83)=0.7734$

The weighing between 77 grams and 83 grams is given by,

$P(77<z<83)=P(z<83)-P(z<77)$

$P(77<z<83)=0.7734-0.2266$

$P(77<z<83)=0.5468$

Converting into percentage,

$P(77<z<83)=0.5468\times 100$

$P(77<z<83)=54.68\%$

Round to nearest whole number - $54.68\%\approx 55\%$

Therefore, The percentage of cockroaches weighing between 77 grams and 83 grams is about 55%.