Chapter 31, Problem 31.37CYS

To determine

To find: The central line for the mean response temperature $\overline{x}$ for each hour’s sample is plotted by $\overline{x}$ chart.

Answer to Problem 31.37CYS

The correct option is Option (b), ${75}^0\text{F}$.

Explanation of Solution

Reason for the correct option:

The population standard deviation and mean are $\sigma =0.5°\text{F}$ and mean $\mu =75°\text{F}$ for the thermostatic. Sample of size of thermostatic is 4 at each hour.

The central line or limit of $\overline{x}$ is the $\mu$, that is the mean temperature of $\mu =75°\text{F}$, which is the temperature at which the thermostats must turn on the air conditioners.

Thus, the option (b) will be the correct option.

Reason for the incorrect answer:

Option (a) $0.5°\text{F}$.

It is the population standard deviation for thermostatic. The value of $\sigma$ is not used in a $\overline{x}$ chart. Thus, the option (a) is incorrect.

Option (c) that is $4°\text{F}$, which is also not same to the mean of temperature of thermostatic. Hence, option (c) is incorrect.

Conclusion:

Thus, the central line for the mean response temperature $\overline{x}$ chart is $75°\text{F}$.

Statistics Concept Introduction

Introduction:

The $\overline{x}$ Chart is a type of variable chart. It is used to monitor the means of the successive sample of constant size.

In this chart there are three types of limits are as follows:

$\begin{array}{c}\text{UCL}=\mu +3\frac{\sigma }{\sqrt{n}}\\ \text{Central}\text{limit}\text{or}\text{line}=\mu \\ \text{LCL}=\mu -3\frac{\sigma }{\sqrt{n}}\end{array}$

Where,

• $\mu$ is the population mean.
• $\sigma$ is the population standard deviation.
• n is the sample mean.