“If all the statisticians in the world were laid head to toe, they wouldn’t be able to reach a conclusion.”
- Anonymous
Class Announcements
Recall: Sampling distribution for means
Theorem: If a variable \(Y\) has a normal distribution in a population, then the distribution of sample means \(\bar{Y}\) is also normal.
Theorem:\(Y \sim N(\mu,\sigma^2) \Rightarrow \bar{Y} \sim N(\mu,\sigma_{\bar{Y}}^2)\), where \(\sigma_{\bar{Y}}\) is the standard error of the mean given by \[
\sigma_{\bar{Y}} = \frac{\sigma}{\sqrt{n}}.
\]
Recall: Sampling distribution for means
We can create a standard normal deviate from the sampling distribution as follows:
\[
Z = \frac{\bar{Y}-\mu}{\sigma_{\bar{Y}}},
\] where \(\sigma_{\bar{Y}} = \frac{\sigma}{\sqrt{n}}\).
Problem: We use \(\mathrm{SE}_{\bar{Y}}\) instead of \(\sigma_{\bar{Y}}\)!!
The Student’s t-distribution
Definition: For \(Y \sim N(\mu,\sigma^2)\), the standard normal deviate\[
Z = \frac{\bar{Y}-\mu}{\sigma_{\bar{Y}}}
\] is normally distributed with mean 0 and standard deviation 1.
Definition: For \(Y \sim N(\mu,\sigma^2)\), the statistic defined by \[
t = \frac{\bar{Y}-\mu}{\mathrm{SE}_{\bar{Y}}}
\] has a Student’s \(t\)-distribution with \(n-1\) degrees of freedom.
The Student’s t-distribution
More probability in the tail for \(t\) since approximating \(\sigma_{\bar{Y}}\) (\(Z\)) by \(\mathrm{SE}_{\bar{Y}}\) (\(t\)) gives us more uncertainty.
What’s with “Student”?
“Guinness brewer William S. Gosset’s work is responsible for inspiring the concept of statistical significance, industrial quality control, efficient design of experiments and, not least of all, consistently great tasting beer.”
– Dan Kopf
“Gosset used a pseudonym [Student] because Guinness prohibited its employees from publishing, following the unauthorized release of some brewing secrets a few years earlier by another employee.”
The variable of interest is normally distributed in the population.
Data are a random sample from population.
Remember, though, that the Central Limit Theorem can make our results somewhat robust to Assumption #1.
Estimation: Confidence interval for mean
Definition: The 95% confidence interval for the mean is given by \[
\overline{Y} - t_{0.05(2),df}\mathrm{SE}_{\overline{Y}} < \mu < \overline{Y} + t_{0.05(2),df}\mathrm{SE}_{\overline{Y}}.
\]
Consider the changes in highest elevation for 31 taxa, in meters, over the late 1900s and early 2000s. Positive and negative numbers will indicate upward and downward shifts in elevation, respectively.
Length Mean Sd Sd err
31 39.32903 30.66312 5.507259
Practice Problem #1
95% confidence interval
(tcrit <-qt(0.025, df=n-1, lower.tail=FALSE))
[1] 2.042272
ci <-c(mu - tcrit*sderr, mu + tcrit*sderr)names(ci) <-c("lower bound", "upper bound")ci
lower bound upper bound
28.08171 50.57635
Practice Problem #1
99% confidence interval
(tcrit <-qt(0.01/2, df=n-1, lower.tail=FALSE))
[1] 2.749996
ci <-c(mu - tcrit*sderr, mu + tcrit*sderr)names(ci) <-c("lower bound", "upper bound")ci
lower bound upper bound
24.18410 54.47397
Hypothesis testing: One-sample t-test
Definition: The one-sample \(t\)-test compares the mean of a random sample from a normal population with the population mean proposed in a null hypothesis.
Null hypothesis: \(H_{0}\): The true mean equals \(\mu_{0}\) (\(\mu = \mu_{0}\)) Alternate hypothesis: \(H_{A}\): The true mean does not equal \(\mu_{0}\) (\(\mu \neq \mu_{0}\)) Test statistic: \[
t = \frac{\overline{Y}-\mu_{0}}{\mathrm{SE}_{\overline{Y}}}
\]Sampling distribution of \(t\) under \(H_{0}\): \(t\)-distribution with \(df = n-1\)
Conclusion: With significance level \(\alpha = 0.05\), since \(P < 0.05\), we reject the null hypothesis.
Practice Problem #1 (Elevation)
One-sample \(t\)-test
\[
H_{0}: \mu = 0 \\
H_{A}: \mu \neq 0
\]
Second, using critical values:
(tstat)
[1] 7.141309
(tcrit <-qt(0.025, df=n-1, lower.tail=FALSE))
[1] 2.042272
Practice Problem #1 (Elevation)
One-sample \(t\)-test
\[
H_{0}: \mu = 0 \\
H_{A}: \mu \neq 0
\]
Third, using t.test function:
t.test(elevation, mu=0, conf.level=0.95)
One Sample t-test
data: elevation
t = 7.1413, df = 30, p-value = 6.057e-08
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
28.08171 50.57635
sample estimates:
mean of x
39.32903
Estimating population variance
In many cases, it isn’t the mean that we are interested in estimating but the variability of a population measure.
Remember, variance is also a population parameter, so we should be able to estimate it.
Stalk-eyed flies have staring contests! Longer stalked flies usually win.
Estimating population variance
Definition: If \(Y\) has a normal distribution, then the sampling distribution of the quantity \[
\chi^{2} = (n-1)s^2/\sigma^2
\] is the \(\chi^2\) distribution with \(n-1\) degrees of freedom.
