What if I want to know the average frequency score for the whole population of beads?
How can we quantify how much we’d expect the mean to differ from one random sample to another?
We need a measure of uncertainty
How about the standard error of the mean?
The standard error is how much we expect the sample mean to vary from one random sample to another.
Standard Error
How can we quantify how much we’d expect the mean to differ from one random sample to another?
\[ \Large\frac{s}{\sqrt{n}}\] . . .
where \(s\) is the sample standard deviation of \(y\).
Standard Error
How can we quantify how much we’d expect the mean to differ from one random sample to another?
\[ \Large\frac{s}{\sqrt{20}}\]
where \(s\) is the sample standard deviation of \(y\).
Sample standard deviation
\[
\Large s = \sqrt{\frac{\sum_{i=1}^n (y_i - \bar{y})^2}{n-1}}
\]
Sample standard deviation
\[
\Large s = \sqrt{\frac{\sum_{i=1}^n (y_i - \bar{y})^2}{20-1}}
\]
confidence intervals
If we use the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter (the average frequency score) to fall within the interval estimates 95% of the time.
Confidence interval
\[\bar{y} \pm t^∗ \times SE_{\bar{y}}\]
\(t^*\) is the critical value for the \(t_{n−1}\) density curve to obtain the desired confidence level
Often we want a 95% confidence level.
Let’s do it in R!
qt(0.025, df =20-1, lower.tail =FALSE)
Let’s do it in R!
Why 0.025?
qt(0.025, df =20-1, lower.tail =FALSE)
Let’s do it in R!
Why lower.tail = FALSE?
qt(0.025, df =20-1, lower.tail =FALSE)
qt(0.975, df =20-1)
confidence intervals
If we use the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter (the mean) to fall within the interval estimates 95% of the time.
Application Exercise
Come draw 20 beads from each of the two groups
Using the frequency score table provided on the handout, write out the data set for each of your samples in the space provided
Calculate the mean, standard deviation, and confidence intervals “by hand” using the \(t^*\) value provided
Make sure to use the standard error in your confidence interval, this is the standard deviation divided \(\sqrt{20}\)
