Question

attempt the following practice exercises. you may reference your notes but not the internet. submit your own individual solutions by the end of class. show your work, in pencil, so that feedback might be provided. collaboration is encouraged, and questions are welcomed!

1. the heights (in feet) of corn plants are normally distributed, with \\( \mu = 7.60 \\) and \\( \sigma = 1.2 \\). if you randomly select 36 corn plants, what is the probability that their mean height is less than 7.3 feet?

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 1.2$ and $n = 36$, we have $\sigma_{\bar{x}}=\frac{1.2}{\sqrt{36}}=\frac{1.2}{6}=0.2$.

Step2: Calculate the z - score

The formula for the z - score is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Here, $\bar{x}=7.3$, $\mu = 7.6$, and $\sigma_{\bar{x}}=0.2$. So $z=\frac{7.3 - 7.6}{0.2}=\frac{- 0.3}{0.2}=-1.5$.

Step3: Find the probability

We want to find $P(\bar{X}<7.3)$, which is equivalent to $P(Z < - 1.5)$. Using the standard normal distribution table, the probability corresponding to $z=-1.5$ is $0.0668$.

Answer:

$0.0668$