Ch. 7 & Ch. 8: Applications of \(t\)-tests

STAT 218 - Week 6, Lecture 3 Lab 4

February 14th, 2024

What Have We Learned So Far?

A Snapshot

  • Descriptive Statistics describes and summarizes data.
    • Includes understanding
      • distribution shape (skewed, normal, etc.),
      • central tendency (mean, median, mode), and
      • spread (variance, standard deviation, range).
  • Inferential Statistics makes predictions and draws conclusions about populations based on sample data.
    • We saw different types of t-tests:
      • One sample t-test: compares a sample mean to a known or hypothesized population mean.
      • Independent samples t-test: compares means of two independent groups.
      • Paired samples t-test: compares means of two related groups or one group with two different measurements/occasions.

Estimation and Hypothesis Testing

We learned that we can estimate the unknown parameters in two ways:

  • Point estimation: A single value calculated from the sample (e.g., \(\bar{y}\))

  • Confidence Intervals: A range of values within which the parameter is expected to fall, with a certain degree of confidence.(e.g., 95% CI, 90% CI)

We also learned that we can use hypothesis testing to test for a specific value(s) of the parameter.

  • e.g., \(H_0: \mu = 76\) cm & \(H_A: \mu \neq 76\) cm (Two-tailed test)
  • \(H_0: \mu_1 - \mu_2 = 0\) cm \(H_A: \mu_1 > \mu_2\) (One-tailed test)

Hypothesis Testing Steps

Revising the Steps of Hypothesis Testing and \(p\)-values

Important

Last lecture, we had 4 steps

  1. Construct the Hypotheses of \(H_0\) and \(H_A\)
  2. Determine your \(\alpha\) level
  3. Calculate test statistic and find the P-value
  4. Draw conclusion.

Revising the Steps of Hypothesis Testing and \(p\)-values

Important

Today, we will add one more step to these lab sessions

  1. Construct the Hypotheses of \(H_0\) and \(H_A\)
  2. Determine your \(\alpha\) level
  3. Check the assumptions
  4. Compute test statistic and find the P-value (Interpreting R Output)
  5. Draw conclusion.

Assumptions - Verification of Conditions

  • It is always important to check first whether the conditions are reasonable in a given case.
  • Here is the list of assumptions that we should be aware of for \(t\)-tests.

1) Random Sampling: the data can be regarded as coming from independently chosen random sample(s),

2) Independence of Observations: the observations should be independent within each sample, and

3) Normal Distribution: Many of the methods depend on the data being from a population that has a normal distribution.

  • REMEMBER! If sample size is large, then condition (3) is less important (Central Limit Theorem).

Normality Assumption - I

  • If the only source of information is the data at hand, then normality can be roughly checked by making a histogram and normal quantile plot of the data.

    • Unfortunately, for a small or moderate sample size, this check is fairly rudimentary.
    • If the sample is large, then the visual plots give us good information about the population shape;
      • However, if \(n\) is large, the requirement of normality is less important anyway due to the Central Limit Theorem.

  • In any case, a rudimentary check is better than none, and every data analysis should begin with inspection of a graph of the data, with special attention to any observations that lie very far from the center of the distribution.

Normality Assumption - II

We check assumptions before conducting any statistical analysis. To check normality assumption, we need to first check sample size.

  • \(1^{st}\) option - small samples: Check the \(p\)-value of Shapiro Wilk test. It is best used with a sample size less than 50 (Shapiro & Wilk 1965; Uttley,2019).

  • \(2^{nd}\) option - large samples: Check the visual plots (e.g., histogram, normal quantile plot) if your sample size is more than 50.

One Sample \(t\)-test

To-Do List

  • Download the worksheet and dataset from the course website.
  • Save them to your STAT 218 folder (VERY IMPORTANT).

Introduction

  • In one sample \(t\) tests, data collected from one sample and compares the mean score with a test value.
    • Test value can be
    1. reported previously in the literature; or
    2. found by calculating level of chance.
  • In this example, we will work with the several data sets. To begin with, we need infer package to conduct t tests.
  • Let’s load that package by using install.packages() function.
install.packages("infer")

Introduction

After loading that package, let’s use library function and load the data sets that we will use today.

library(tidyverse)
library(infer)
library(palmerpenguins)
data("penguins") 
stream_data <- read_csv("Example 8.3.4.csv")

Example of a Case:

Imagine that you are a biologist studying penguins, particularly their bill lengths. You hypothesize that the average bill length of penguins is 40 mm and you collect a random sample of 344 penguins, measure and record their bill length in mm.

Perform a one sample \(t\)-test to investigate whether the bill length of the penguins differs from the test value of 40 mm. Use the 5% significance level (\(\alpha = 0.05\)).

Checking the Normality Assumption

  • Check your sample size first!
    • \(n\) > 50, so we can check visual plots.

Quantile Plot

ggplot(data = na.omit(penguins),
       aes(sample = bill_length_mm)) +
  geom_qq()

Interpreting the Output

t_test(penguins, response = bill_length_mm, mu = 40)
# A tibble: 1 × 7
  statistic  t_df  p_value alternative estimate lower_ci upper_ci
      <dbl> <dbl>    <dbl> <chr>          <dbl>    <dbl>    <dbl>
1      13.3   341 9.58e-33 two.sided       43.9     43.3     44.5

Conclusion: As \(P\)-value is very small, we can reject \(H_0\) and conclude that our data provided sufficient evidence to support the claim that the bill length of the penguins differs from 40 mm.

Confidence Interval: Type your confidence interval statement to your worksheet!

Independent Samples \(t\)-test

Introduction

  • Independent samples \(t\) test compares means of two independent groups.
  • 1 DV (numeric) + 1 IV(categorical)
    • Your IV should have 2 groups (e.g., male, female)

Example of a Case:

Now, you’re curious about the difference in the body mass of penguins based on their sex. You hypothesize that body mass varies between different sexes. To test your hypothesis, you collect a random sample of 344 penguins, measure their body mass, and record their sex.

Perform an independent samples \(t\)-test to investigate whether the body mass of penguins differs between different sexes. Use the 10% significance level (\(\alpha = 0.10\)).

Checking the Normality Assumption

  • Check your sample size first!
    • \(n\) > 50, so we can check visual plots.

Quantile Plots

ggplot(data = na.omit(penguins),
       aes(sample = body_mass_g)) +
  geom_qq() +
  facet_wrap(~sex)

Interpreting the Output

t_test(x = penguins, 
       formula = body_mass_g ~ sex, 
       order = c("male", "female"),
       alternative = "two-sided",
       conf_level = 0.90)
# A tibble: 1 × 7
  statistic  t_df  p_value alternative estimate lower_ci upper_ci
      <dbl> <dbl>    <dbl> <chr>          <dbl>    <dbl>    <dbl>
1      8.55  324. 4.79e-16 two.sided       683.     552.     815.

Conclusion: Type your conclusion statement to your worksheet!

Confidence Interval: Type your confidence interval statement to your worksheet!

Paired Samples \(t\)-test

Introduction

  • Paired samples \(t\) test compares means of two related groups or one group with two different measurements/occasions.
  • 1 DV (numeric) + 1 IV(categorical)
    • Your IV should be two related groups or one group with two different measurements/occasions.

Example of a Case: Pollutants in a stream may accumulate or attenuate as water flows down the stream. In a study to monitor the accumulation and attenuation of fecal contamination in a stream running through cattle rangeland, monthly water specimens were collected at two locations along the stream over a period of 21 months.

The data set stream the total coliform count (MPN/100ml) for a water specimen.

Perform a paired samples \(t\)-test to assess whether the mean total coliform count is consistent across the two locations. Use the 5% significance level (\(\alpha = 0.05\)).

Checking the Normality Assumption

  • Check your sample size first!
    • \(n\) < 50, so we should check Shapiro-Wilk test.
  • This output is from a calculation of Shapiro-Wilk test. We generally use Shapiro-Wilk test for relatively smaller sample size because visuals can be misleading in smaller sample sizes. Please interpret Shapiro-Wilk \(p\)-value.

Shapiro-Wilk normality test

data: stream$Difference W = 0.9641, p-value = 0.6022

Interpreting the Output

# A tibble: 1 × 7
  statistic  t_df  p_value alternative estimate lower_ci upper_ci
      <dbl> <dbl>    <dbl> <chr>          <dbl>    <dbl>    <dbl>
1      4.61    20 0.000170 two.sided      1103.     604.    1602.

Conclusion: Type your conclusion statement to your worksheet!

Confidence Interval: Type your confidence interval statement to your worksheet!