Respuesta :
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,
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Where, x is the sample mean
[tex]\mu=80[/tex] is the population mean
[tex]\sigma=4[/tex] is the standard deviation.
For x=77 z-score is
[tex]z=\frac{77-80}{4}[/tex]
[tex]z=\frac{-3}{4}[/tex]
[tex]z=-0.75[/tex]
For x=83 z-score is
[tex]z=\frac{83-80}{4}[/tex]
[tex]z=\frac{3}{4}[/tex]
[tex]z=0.75[/tex]
The probability of z lie between 77 and 83 is given by z-score table,
[tex]P(z<77)=0.2266[/tex]
[tex]P(z<83)=0.7734[/tex]
The weighing between 77 grams and 83 grams is given by,
[tex]P(77<z<83)=P(z<83)-P(z<77)[/tex]
[tex]P(77<z<83)=0.7734-0.2266[/tex]
[tex]P(77<z<83)=0.5468[/tex]
Converting into percentage,
[tex]P(77<z<83)=0.5468\times 100[/tex]
[tex]P(77<z<83)=54.68\%[/tex]
Round to nearest whole number - [tex]54.68\%\approx 55\%[/tex]
Therefore, The percentage of cockroaches weighing between 77 grams and 83 grams is about 55%.
