Question

a machine is set to fill milk containers with a mean of 63 ounces and a standard deviation of 0.11 ounce. a random sample of 39 containers has a mean of 63.04 ounces. the machine needs to be reset when the mean of a random sample is unusual. does the machine need to be reset? explain. because the z - score () within the range of a usual event, namely within of the mean of the sample means. (round to two decimal places as needed.)

Explanation:

Step1: Identify the formula for the z - score of a sample mean

The formula for the z - score of a sample mean is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Step2: Substitute the given values into the formula

We are given that $\mu = 63$, $\sigma=0.11$, $n = 39$, and $\bar{x}=63.04$.
$z=\frac{63.04 - 63}{\frac{0.11}{\sqrt{39}}}$
First, calculate the denominator: $\sqrt{39}\approx6.245$, $\frac{0.11}{6.245}\approx0.0176$.
Then, calculate the numerator: $63.04 - 63=0.04$.
So, $z=\frac{0.04}{0.0176}\approx2.27$.

Step3: Determine if the z - score is within the range of a usual event

Usual events have z - scores within the range of $- 2$ to $2$. Since $z = 2.27>2$, the sample mean is unusual.

Answer:

Yes, because the z - score ($2.27$) is not within the range of a usual event, namely within $2$ of the mean of the sample means.