Question

A fast-food restaurant uses an automated filling machine to pour its soft drinks. The machine has different settings for small, medium, and large drink cups. According to the machine’s manufacturer, when the large setting is chosen, the amount of liquid dispensed by the machine follows a Normal distribution with mean $27$ounces and standard deviation $0.8$ounces. When the medium setting is chosen, the amount of liquid dispensed follows a Normal distribution with mean $17$ounces and standard deviation $0.5$ounces. To test the manufacturer’s claim, the restaurant manager measures the amount of liquid in a random sample of $25$cups filled with the medium setting and a separate random sample of $20$cups filled with the large setting. Let ${\overline{x}}_1-{\overline{x}}_2$be the difference in the sample mean amount of liquid under the two settings (large – medium). What is the shape of the sampling distribution of${\overline{x}}_1-{\overline{x}}_2$. Why?

Short answer

It is approximately normally distributed.

Step-by-step solution

Given information

Sample mean$=27$

Sample mean$=17$

Standard deviationrole="math" localid="1650291670593" $=0.8$

Standard deviation$=0.5$

Explanation

Given is written as

${\overline{x}}_1=27$

${\overline{x}}_2=17$

${\sigma }_1=0.8$

${\sigma }_2=0.5$

It has been assumed that population distributions are normal. As a result, the sample distribution of ${\overline{x}}_1-{\overline{x}}_2$can be considered to be approximately normally distributed.