the heights of a species of plant are approximately normally distributed, have a mean of 9.31 cm, and have a…

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard - deviation and $n$ is the sample size. Given $\sigma = 0.55$ cm and $n = 20$, then $\sigma_{\bar{x}}=\frac{0.55}{\sqrt{20}}\approx\frac{0.55}{4.472}\approx0.123$.

Step2: Calculate the z - score

The z - score formula for the sample mean is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, and $\sigma_{\bar{x}}$ is the standard error of the mean. Given $\bar{x}=9.5$ cm, $\mu = 9.31$ cm, and $\sigma_{\bar{x}}\approx0.123$ cm. Then $z=\frac{9.5 - 9.31}{0.123}=\frac{0.19}{0.123}\approx1.54$.

Step3: Find the probability

We want to find $P(\bar{X}<9.5)$, which is equivalent to $P(Z < 1.54)$ when using the standard normal distribution. Looking up the value of $P(Z < 1.54)$ in the standard - normal table, we get $P(Z < 1.54)=0.9382\approx0.9388$.

Answer:

0.9388