Question

the average birth weight of domestic cats is about 4 ounces. assume that the distribution of birth weights is normal with a standard deviation of 0.5 ounce.
a. find the birth weight of cats at the 70th percentile.
b. find the birth weight of cats at the 30th percentile.

a. the birth weight of cats at the 70th percentile is ounces.
(round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for the z - score and percentile relationship

For a normal distribution \(X\sim N(\mu,\sigma)\), the formula to find the value \(x\) is \(x = \mu+z\sigma\), where \(\mu = 4\) (mean), \(\sigma=0.5\) (standard deviation). We need to find the \(z\) - score corresponding to the given percentile.
We use the standard normal distribution table (or a calculator with a normal - distribution function).

Step2: Find the \(z\) - score for the 70th percentile

Using a standard normal table or a calculator (e.g., in R: qnorm(0.7,0,1) or in Excel: NORM.S.INV(0.7)), the \(z\) - score \(z_{70}\) corresponding to a cumulative probability of \(0.7\) is approximately \(z = 0.52\).
Then, using the formula \(x=\mu + z\sigma\), substitute \(\mu = 4\), \(\sigma = 0.5\), and \(z = 0.52\)

$$x=4+0.52\times0.5$$

$$x = 4 + 0.26$$

$$x=4.26$$

Answer:

\(4.26\)