Question

A machine is designed to fill 16-ounce bottles of shampoo. When the machine is working properly, the amount poured into the bottles follows a normal distribution with mean 16.05 ounces with a standard deviation of .2005 ounces. If four bottles are randomly selected each hour and the number of ounces in each bottle is measured, then 95% of the means calculated should occur in what interval? Hint: the standard deviation rule says that 95% of the observations are within how many standard deviations away from the mean? Round answers to four decimal places. ( , )

Answer 1

Answer:

(15.8495,16.2505)  

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 16.05 ounces

Standard Deviation, σ = 0.2005 ounces

We are given that the distribution of amount poured into the bottles is a bell shaped distribution that is a normal distribution.

Empirical rule:

• It states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ).

1. It shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).

Confidence interval:

$\mu \pm 2\frac{\sigma}{\sqrt{n}}$

Putting the values, we get,

$16.05 \pm 2(\frac{0.2005}{\sqrt{4}} ) = 16.05 \pm 0.2005 = (15.8495,16.2505)$