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 4.26 ounces. (round to two decimal places as needed.) b. the birth weight of cats at the 30th percentile is □ ounces. (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for normal distribution

The formula for a value \(x\) in a normal distribution is \(x=\mu + z\sigma\), where \(\mu\) is the mean, \(\sigma\) is the standard deviation, and \(z\) is the z - score. Given \(\mu = 4\) ounces and \(\sigma=0.5\) ounce.

Step2: Find the z - score for the 30th percentile

Using a standard normal table (or a calculator with a normal - distribution function, such as the inverse of the cumulative distribution function for the standard normal distribution), for the 30th percentile (\(p = 0.30\)), the z - score \(z\approx - 0.52\)

Step3: Calculate the birth weight

Substitute \(\mu = 4\), \(\sigma = 0.5\), and \(z=-0.52\) into the formula \(x=\mu+z\sigma\).

$$x = 4+(- 0.52)\times0.5$$

$$x=4 - 0.26$$

$$x = 3.74$$

Answer:

3.74