10.5 Hypothesis Testing

In this section, we introduce the basics of hypothesis testing in R. Hypothesis testing is a fundamental tool in statistics that allows us to make inferences about population parameters based on sample data.

10.5.1 The t-test

The t-test is used to test whether the mean of a population (or the difference between two population means) equals a specified value. R provides the t.test() function for this purpose.

10.5.1.1 One-sample t-test

Suppose we want to test whether the average sale price of houses in the sahp dataset is different from $200,000 (i.e., 200 in the unit of thousands).

library(r02pro)
t.test(sahp$sale_price, mu = 200)
#> 
#>  One Sample t-test
#> 
#> data:  sahp$sale_price
#> t = -3.0911, df = 163, p-value = 0.002346
#> alternative hypothesis: true mean is not equal to 200
#> 95 percent confidence interval:
#>  167.0439 192.7363
#> sample estimates:
#> mean of x 
#>  179.8901

The output includes:

  • t: the test statistic
  • df: degrees of freedom
  • p-value: the probability of observing a test statistic as extreme as the one computed, assuming the null hypothesis is true
  • 95 percent confidence interval: a range of plausible values for the true mean
  • sample estimates: the sample mean

If the p-value is less than our significance level (commonly 0.05), we reject the null hypothesis.

10.5.1.2 Two-sample t-test

We can also compare the means of two groups. Let’s test whether one-story and two-story houses have different average sale prices.

price_1story <- sahp$sale_price[sahp$house_style == "1Story"]
price_2story <- sahp$sale_price[sahp$house_style == "2Story"]
t.test(price_1story, price_2story)
#> 
#>  Welch Two Sample t-test
#> 
#> data:  price_1story and price_2story
#> t = -0.87927, df = 88.388, p-value = 0.3816
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -48.19646  18.62819
#> sample estimates:
#> mean of x mean of y 
#>  183.0277  197.8118

By default, t.test() performs a Welch two-sample t-test, which does not assume equal variances. To assume equal variances, set var.equal = TRUE.

t.test(price_1story, price_2story, var.equal = TRUE)
#> 
#>  Two Sample t-test
#> 
#> data:  price_1story and price_2story
#> t = -0.91686, df = 128, p-value = 0.3609
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -46.68985  17.12157
#> sample estimates:
#> mean of x mean of y 
#>  183.0277  197.8118

10.5.1.3 Paired t-test

A paired t-test is used when observations come in pairs (e.g., before/after measurements). You can specify paired = TRUE in t.test().

10.5.2 The Chi-Squared Test

The chi-squared test is used to test the association between two categorical variables. R provides chisq.test() for this purpose.

Let’s test whether kitchen quality (kit_qual) and house style (house_style) are independent in the sahp dataset.

# Create a contingency table
quality_table <- table(sahp$kit_qual, sahp$house_style)
quality_table
#>            
#>             1.5Fin 1Story 2Story SFoyer SLvl
#>   Average       14     43     18      4    6
#>   Excellent      0      9      5      0    0
#>   Fair           2      5      2      0    0
#>   Good           5     24     25      1    2

# Perform chi-squared test
chisq.test(quality_table)
#> 
#>  Pearson's Chi-squared test
#> 
#> data:  quality_table
#> X-squared = 15.492, df = 12, p-value = 0.2156

A small p-value suggests that the two variables are not independent.

10.5.3 The Correlation Test

To test whether the correlation between two numeric variables is significantly different from zero, use cor.test().

cor.test(sahp$liv_area, sahp$sale_price)
#> 
#>  Pearson's product-moment correlation
#> 
#> data:  sahp$liv_area and sahp$sale_price
#> t = 12.443, df = 162, p-value < 2.2e-16
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  0.6113017 0.7698383
#> sample estimates:
#>       cor 
#> 0.6990621

The output shows the estimated correlation, the test statistic, and the p-value.

10.5.4 Interpreting Results

When interpreting hypothesis test results, keep these points in mind:

  1. A small p-value (typically < 0.05) provides evidence against the null hypothesis, but does not prove that the alternative is true.
  2. Statistical significance does not imply practical significance. A very large sample can make tiny differences statistically significant.
  3. The confidence interval provides more information than the p-value alone, as it shows the range of plausible values for the parameter.

10.5.5 Exercises

  1. Using the gm2004 dataset from the r02pro package, perform a one-sample t-test to test whether the average life_expectancy across all countries is different from 70. What do you conclude at the 0.05 significance level?

  2. Using the gm2004 dataset, perform a two-sample t-test to compare the average cholesterol between "male" and "female" groups (the gender variable). Is there a statistically significant difference?

  3. Create a contingency table of continent and gender from the gm2004 dataset. Perform a chi-squared test to determine whether continent and gender are independent.

  4. Test whether the correlation between BMI and cholesterol in the gm2004 dataset is statistically significant. Report the estimated correlation and the p-value.

  5. Using the sahp dataset, test whether houses with central air conditioning (central_air == "Y") have a significantly higher average sale_price than those without. Use a two-sample t-test with var.equal = FALSE.

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