Ten samples of n = 5 were collected to construct an $\overline { x ^ { * } }$ chart. The sample mean and range for each sample are shown in the table below. $$\begin{array} { | c c c | c c c | }
\hline \text { Sample } & \text { Mean } & \text { Range } & \text { Sample } & \text { Mean } & \text { Range } \
\hline 1 & 215 & 15 & 6 & 224 & 21 \
2 & 204 & 20 & 7 & 205 & 27 \
3 & 160 & 10 & 8 & 184 & 12 \
4 & 226 & 19 & 9 & 207 & 16 \
5 & 232 & 24 & 10 & 212 & 17 \
\hline
\end{array}$$ Calculate the empirical centerline for the $\overline { x ^ { * } }$ chart.
A)210.5
B)206.9
C)205.3
D)208.2

Question

Ten samples of n = 5 were collected to construct an $\overline { x ^ { * } }$ chart. The sample mean and range for each sample are shown in the table below. $$\begin{array} { | c c c | c c c | } 
\hline \text { Sample } & \text { Mean } & \text { Range } & \text { Sample } & \text { Mean } & \text { Range } \
\hline 1 & 215 & 15 & 6 & 224 & 21 \
2 & 204 & 20 & 7 & 205 & 27 \
3 & 160 & 10 & 8 & 184 & 12 \
4 & 226 & 19 & 9 & 207 & 16 \
5 & 232 & 24 & 10 & 212 & 17 \
\hline
\end{array}$$ Calculate the empirical centerline for the $\overline { x ^ { * } }$ chart.
A)210.5
B)206.9
C)205.3
D)208.2

Quizplus · Accepted Answer

The empirical centerline for the $\overline{x^*}$ chart is calculated using the formula:
$$\overline{x} = \frac{1}{n}\sum_{i=1}^{k} \overline{x_i}$$
where $\overline{x_i}$ is the sample mean for the $i$th sample and $k$ is the number of samples, in this case, $k=10$.
Substituting the values given in the table, we get:
$$\overline{x} = \frac{1}{10}\sum_{i=1}^{10} \overline{x_i} = \frac{215+204+160+226+232+224+205+184+207+212}{10} = 206.9$$
Therefore, the empirical centerline is 206.9, which corresponds to choice (B).

Ten Samples of N = 5 Were Collected to Construct x∗‾\overline { x ^ { * } }x∗

Ten Samples of N = 5 Were Collected to Construct $\overline { x ^ { * } }$