• We can't construct sampling distributions directly, because we don't have access to the entire population data
  • this is the whole point of statistical inference: observe only one sample, try to make inference about the entire population
• Hence we rely on the Central Limit Theorem to tell us what the sampling distribution would look like, if we could construct it

If certain conditions are met, the sampling distribution of the sample statistic will be nearly normally distributed with mean equal to the population parameter and standard error equal inversely proportional to the sample size.

• Single mean: \(\bar{x} \sim N\left(mean = \mu, SE = \frac{\sigma}{\sqrt{n}}\right)\)
• Difference between two means: \((\bar{x}_1 - \bar{x}_2) \sim N\left(mean = (\mu_1 - \mu_2), SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \right)\)
• Single proportion: \(\hat{p} \sim N\left(mean = p, SE = \sqrt{\frac{p (1-p)}{n}} \right)\)
• Difference between two proportions: \((\hat{p}_1 - \hat{p}_2) \sim N\left(mean = (p_1 - p_2), SE = \sqrt{\frac{p_1 (1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \right)\)

• Independence: The sampled observations must be independent. This is difficult to check, but the following are useful guidelines:
  • the sample must be random
  • if sampling without replacement, sample size must be less than 10% of population size
• Sample size / distribution:
  • numerical data: The more skewed the sample (and hence the population) distribution, the larger samples we need. Usually n > 30 is considered a large enough sample for population distributions that are not extremely skewed.
  • categorical data: At least 10 successes and 10 failures.
• If comparing two populations, the samples must be independent of each other, and all conditions should be checked for both groups.

If necessary conditions are met, we can also use inference methods based on the CLT:

• use the CLT to calculate the SE of the sample statistic of interest (sample mean, sample proportion, difference between sample means, etc.)

• calculate the test statistic, number of standard errors away from the null value the observed sample statistic is
  • T for means, along with appropriate degrees of freedom
  • Z for proportions

• use the test statistic to calculte the p-value, the probability of an observed or more extreme outcome given that the null hypothesis is true

• Also unimodal and symmetric, and centered at 0

• Thicker tails than the normal distribution (to make up for additional variability introduced by using \(s\) instead of \(\sigma\) in calculation of the SE)

• Parameter: degrees of freedom
  • df for single mean: \(df = n - 1\)
  • df for comparing two means: \(df = min(n_1 - 1, n_2 - 1)\)

• Finding probabilities under the t curve:

pt(-1.96, df = 9)

## [1] 0.0408222

pt(1.96, df = 9, lower.tail = FALSE)

## [1] 0.0408222

• Finding cutoff values under the normal curve:

qt(0.025, df = 9)

## [1] -2.262157

qt(0.975, df = 9)

## [1] 2.262157

• Since 1972, the General Social Survey (GSS) has been monitoring societal change and studying the growing complexity of American society.

• The GSS aims to gather data on contemporary American society in order to
  • monitor and explain trends and constants in attitudes, behaviors, attributes;
  • examine the structure and functioning of society in general as well as the role played by relevant subgroups;
  • compare the US to other societies to place American society in comparative perspective and develop cross-national models of human society;
  • make high-quality data easily accessible to scholars, students, policy makers, and others, with minimal cost and waiting.

• GSS questions cover a diverse range of issues including national spending priorities, marijuana use, crime and punishment, race relations, quality of life, confidence in institutions, and sexual behavior.

2010 GSS:

gss <- read.csv("https://stat.duke.edu/~mc301/data/gss2010.csv")

• Data dictionary at https://gssdataexplorer.norc.org/variables/vfilter

• Note that not all questions are asked every year

p-value = P(observed or more extreme outcome | \(H_0\) true)

pt(t, df, lower.tail = FALSE)

## [1] 2.720895e-18

• Since the p-value is small, we reject \(H_0\).

• The data provide convincing evidence that Americans, on average, spend more than 3 hours per day relaxing after work.

\[point~estimate \pm critical~value \times SE\]

t_star <- qt(0.95, df)
pt_est <- hrsrelax_summ$xbar
round(pt_est + c(-1,1) * t_star * se, 2)

## [1] 3.55 3.81

• Two sided alternative HT with \(\alpha\) \(\rightarrow\) \(CL = 1 - \alpha\)
• One sided alternative HT with \(\alpha\) \(\rightarrow\) \(CL = 1 - (2 times \alpha)\)

hrsrelax_sex_summ <- gss %>% 
  filter(!is.na(hrsrelax)) %>%
  group_by(sex) %>%
  summarise(xbar = mean(hrsrelax), s = sd(hrsrelax), n = length(hrsrelax))
hrsrelax_sex_summ

## Source: local data frame [2 x 4]
## 
##      sex     xbar        s     n
##   (fctr)    (dbl)    (dbl) (int)
## 1 FEMALE 3.449180 2.396948   610
## 2   MALE 3.939338 2.848216   544

p-value = P(observed or more extreme outcome | \(H_0\) true)

pt(t, df) * 2

## [1] 0.001767347

\[point~estimate \pm critical~value \times SE\]

t_star <- qt(0.975, df)
pt_est <- hrsrelax_sex_summ$xbar[1] - hrsrelax_sex_summ$xbar[2]
round(pt_est + c(-1,1) * t_star * se, 2)

## [1] -0.80 -0.18