The Central Limit Theorem

For a population with a well-defined mean $\mu$ and standard deviation $\sigma$, these three properties hold for the distribution of sample average $\overline{X}$, assuming certain conditions hold:

✅ The distribution of the sample statistic is nearly normal

✅ The distribution is centered at the (often unknown) population parameter

✅ The variability of the distribution is inversely proportional to the square root of the sample size

Why do we care?

Knowing the distribution of the sample statistic $\overline{X}$ can help us

• estimate a population parameter as point estimate $\pm$ margin of error
  • the margin of error is comprised of a measure of how confident we want to be and how variable the sample statistic is

• test for a population parameter by evaluating how likely it is to obtain to observed sample statistic when assuming that the null hypothesis is true
  • this probability will depend on how variable the sampling distribution is

Inference based on the CLT

If necessary conditions are met, we can also use inference methods based on the CLT. Suppose we know the true population standard deviation, $\sigma$.

Then the CLT tells us that $\overline{X}$ approximately has the distribution $N(\mu ,\sigma /\sqrt{n})$.

That is,

$$Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}\sim N(0,1)$$

T distribution

In practice, we never know the true value of $\sigma$, and so we estimate it from our data with $s$.

We can make the following test statistic for testing a single sample's population mean, which has a t-distribution with n-1 degrees of freedom:

T distribution

• The t-distribution is also unimodal and symmetric, and is centered at 0

• It has thicker tails than the normal distribution

  • This is to make up for additional variability introduced by using $s$ instead of $\sigma$ in calculation of the SE

• It is defined by the degrees of freedom

T vs Z distributions

T distribution

#P(t < -1.96)
pt(-1.96, df = 9)

## [1] 0.0408222

#P(t > -1.96)
pt(-1.96, df = 9, 
   lower.tail = FALSE)

## [1] 0.9591778

# Find Q1
qt(0.25, df = 9)

## [1] -0.7027221

# Q3
qt(0.75, df = 9)

## [1] 0.7027221

Resident satisfaction in Durham

durham_survey contains resident responses to a survey given by the City of Durham in 2018. These are a randomly selected, representative sample of Durham residents.

Questions were rated 1 - 5, with 1 being "highly dissatisfied" and 5 being "highly satisfied."

Exploratory Data Analysis

durham <- read_csv("data/durham_survey.csv") %>%
  filter(quality_library != 9)

durham %>% 
  summarise(x_bar = mean(quality_library), 
            med = median(quality_library), 
            sd = sd(quality_library), 
            n = n())

## # A tibble: 1 × 4
##   x_bar   med    sd     n
##   <dbl> <dbl> <dbl> <int>
## 1  3.97     4 0.900   521

Hypotheses

$$H_0:\mu =3$$ $$H_a:\mu >3$$

Conditions

Independence?

✅ The residents were randomly selected for the survey, and 521 is less than 10% of the Durham population (~ 270,000).

Sample size / distribution?

✅ 521 > 30, so the sample is large enough to apply the Central Limit Theorem.

Calculating the test statistic

Summary statistics from the sample:

## # A tibble: 1 × 3
##    xbar     s     n
##   <dbl> <dbl> <int>
## 1  3.97 0.900   521

And the CLT says:

$$\overline{x}\sim N(mean=\mu ,SE=\frac{\sigma }{\sqrt{n}})$$

Calculating the test statistic

(se <- durham_summary$s / sqrt(durham_summary$n)) # SE

## [1] 0.03944416

(t <- (durham_summary$xbar - 3) / se) # Test statistic

## [1] 24.57372

Calculating the p-value

(df <- durham_summary$n - 1) # Degrees of freedom

## [1] 520

pt(t, df, lower.tail = FALSE) # P-value, P(T > t |H_0 true)

## [1] 2.247911e-89

Conclusion

The p-value is very small, so we reject $H_0$.

The data provide sufficient evidence at the $\alpha =0.05$ level that Durham residents, on average, are satisfied with the quality of the public library system $(\mu >3)$

Confidence interval for a mean

Calculate 95% confidence interval

# Critical value 
t_star <- qt(0.975, df)

# Point estimate 
point_est <- durham_summary$xbar

# Confidence interval
CI <- point_est + c(-1,1) * t_star * se
round(CI, 2)

## [1] 3.89 4.05

Interpret 95% confidence interval

The 95% confidence interval is 3.89 to 4.05.

We are 95% confident that the true mean rating for Durham residents' satisfaction with the library system is between 3.89 and 4.05.

CLT-based hypothesis testing in infer

$$H_0:\mu =3\text{vs}{H}_a:\mu >3$$

durham %>%
  t_test(response = quality_library, 
         mu = 3, 
         alternative = "greater", 
         conf_int = FALSE)

## # A tibble: 1 × 5
##   statistic  t_df  p_value alternative estimate
##       <dbl> <dbl>    <dbl> <chr>          <dbl>
## 1      24.6   520 2.25e-89 greater         3.97

CLT-based confidence intervals in infer

Calculate a 95% confidence interval for the mean satisfaction rating.

durham %>%
  t_test(response = quality_library, 
         alternative = "two-sided",
         conf_int = TRUE, conf_level = 0.95)

## # A tibble: 1 × 7
##   statistic  t_df p_value alternative estimate lower_ci upper_ci
##       <dbl> <dbl>   <dbl> <chr>          <dbl>    <dbl>    <dbl>
## 1      101.   520       0 two.sided       3.97     3.89     4.05