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

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

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=(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=(77-80)/(4)`

`z=(-3)/(4)`

`z=-0.75`

For x=83 z-score is

`z=(83-80)/(4)`

`z=(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* 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%.