STAT 1010 - Fall 2022
\[ SD(\bar{X}) = SE(\bar{X}) = \frac{\sigma}{\sqrt{n}} \]
If \(X \sim N(\mu_X, \sigma_X^2)\)
\[ \bar{X} \sim N(\mu = \mu_X, \sigma^2 = \frac{\sigma_X^2}{n}) \]
If \(\bar{X} \sim N(\mu = 12, \sigma^2 = 2.3)\) how can we find the control limits?
pnorm(-1.318761)
\(\approx 0.09362451\)pnorm(10, mean = 12, sd = sqrt(2.3))
If \(\bar{X} \sim N(\mu = 12, \sigma^2 = 2.3)\) how can we find the control limits?
1 - pnorm(1.318761)
\(\approx 0.09362451\)1 - pnorm(14, mean = 12, sd = sqrt(2.3))
qnorm(0.0125)
\(\approx -2.241403\)qnorm(0.0125, mean = 12, sd = sqrt(2.3))
qnorm(1- 0.0125)
\(\approx 2.241403\)qnorm(1-0.0125, mean = 12, sd = sqrt(2.3))
We are not testing once, but multiple times. Assuming independence:
\(\begin{aligned}P(\textrm{within limits for 10 days}) =& P(\textrm{within limits for day 1}) \cdot P(\textrm{within limits for day 2}) \cdot \dots \cdot P(\textrm{within limits for day 10})\\ &= 0.975^{10} \approx 0.7763296 \end{aligned}\)
There is a \(1-0.7763296 = 0.2236704\) percent false positive rate
Management must decide if there is a false positive by checking for mechanical errors and inspecting equipment
Adjust \(\alpha\) value to address this
X-bar charts are slow to detect under or over filling