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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.

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Answer:

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%.

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