Lecture 13?
NC State University
ST 511 - Fall 2024
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– Are you keeping up with Slack?
– Quiz-6 Wednesday (due Sunday)
– Exam-1: October 9th (in-class)
– Exam-1: Assigned October 9th; Due 11:59pm October 15th
– All videos are live
– All solutions are posted
– Exam equation sheet is posted
Notation check
– \(\mu\)
– \(\pi\)
– \(s\)
– \(\hat{p}\)
– \(\bar{x}\)
– \(\sigma\)
– \(\mu\) = population mean
– \(\pi\) = population proportion
– \(s\) = sample standard deviation
– \(\hat{p}\) sample proportion
– \(\bar{x}\) sample mean
– \(\sigma\) population standard deviation
What is the difference between a population distribution and sampling distribution?
A population distribution is a distribution of all observational units of interest, while a sampling distribution is a distribution of a statistic that comes from a random sample of a population
Null distribution (sampling distribution under the assumption of the null hypothesis), and the connection to a named distribution (Z and t).
Howling Cow Example
\(H_o:\) \(\pi\) = .5
\(H_a:\) \(\pi\) < .5
\(\hat{p}\) = \(\frac{37}{100}\)
Our entire goal is to make a null distribution! This distribution is:
– centered at \(\pi_o\)
– has spread (standard error for the null) of
\[ \sqrt{\frac{.5 * (1-.5)}{100}} \]
It has this standard error because we checked our assumptions!
Here is the approximated null distribution. And we can calculate the p-value straight from here!
Z = \(\frac{\hat{p} - \pi_o}{SE}\)
Z = \(\frac{.37 - .5}{0.05}\) = -2.61
When do we make confidence intervals? What question are we trying to answer?
What is the population mean bill length (mm) of penguins?
Includes measurements for penguin species, island in Palmer Archipelago. There are 342 penguins in the data set. For this example, you can assume that one penguin does not influence another penguin, or that each penguin is independent from each other.
What is the other assumption we need to check? How do we check this?
Still a bit subjective (as we learned in hypothesis testing). Stick to the general < 30, > 30, > 60 rules, and we will continue practicing in different scenarios.
\(\bar{x} \pm \text{Margin of Error}\)
Let’s make a 95% confidence interval!
\(\bar{x} \pm t^* * SE\)
\(\bar{x} \pm t^* * \frac{s}{\sqrt{n}}\)
Let’s make a 95% confidence interval!
penguins |>
summarize(center = mean(bill_length_mm, na.rm = T),
spread = sd(bill_length_mm, na.rm = T),
count = n()-2) # subtract 2 NA values out# A tibble: 1 × 3
center spread count
<dbl> <dbl> <dbl>
1 43.9 5.46 342[1] 1.966945Hmmm.. that t* looks familiar…
Let’s make a 95% confidence interval!
\(43.9 \pm 1.97 * \frac{5.46}{\sqrt{342}} = (43.318, 44.482)\)
We are 95% confident that the true mean bill length for penguins on the Palmer island to be between 43.318 and 44.482 mm.
Sample with replacement 342 times
Calculate the mean of your new resampled data
Plot it
Do this process many many times!
