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:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 80, \sigma = 4$

The percentage of cockroaches weighing between 77 grams and 83 grams

This is the pvalue of Z when X = 83 subtracted by the pvalue of Z when X = 83. So

X = 83

$Z = \frac{X - \mu}{\sigma}$

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

$Z = 0.75$

$Z = 0.75$ has a pvalue of 0.7734

X = 77

$Z = \frac{X - \mu}{\sigma}$

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

$Z = -0.75$

$Z = -0.75$ has a pvalue of 0.2266

0.7734 - 0.2266 = 0.5468

Rounded to the nearest whole number, 55%

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