DA_DIS#10_HypothesisTesting

*Note: This first part is taken as notes for myself, not required for this week’s discussion.

Part I Degrees of freedom

• What is it?

  • The degrees of freedom in a statistical calculation represent how many values involved in a calculation have the freedom to vary. 

• Formula

  • DF = N - 1

• Why subtract 1?

  • Link 1 & Link 2

  • As degrees of freedom is about how many independent variables can be freely choose. For example using mean of average, say a group of 25 numbers has an average of 18. Once the first 24 numbers are randomly selected, the value of the last number will be determined by those first 24 numbers randomly selected. Therefore, it is not a factor that can be freely chosen to be any value. Hence, need to be subtracted from the total sample size.

Part II Normal Distribution Plots

# Set up a sequence of values for the x-axis
x_values <- seq(-4, 4, 0.01)

# Create data frames for normal and t-distributions with different degrees of freedom
normal_data <- data.frame(x = x_values, y = dnorm(x_values))
t_dist_df2 <- data.frame(x = x_values, y = dt(x_values, df = 2))
t_dist_df5 <- data.frame(x = x_values, y = dt(x_values, df = 5))
t_dist_df15 <- data.frame(x = x_values, y = dt(x_values, df = 15))
t_dist_df30 <- data.frame(x = x_values, y = dt(x_values, df = 30))
t_dist_df120 <- data.frame(x = x_values, y = dt(x_values, df = 120))

# Combine data frames
combined_data <- bind_rows(
  data.frame(distribution = "Normal", normal_data),
  data.frame(distribution = "t (df=2)", t_dist_df2),
  data.frame(distribution = "t (df = 5)", t_dist_df5), 
  data.frame(distribution = "t (df = 15)", t_dist_df15),
  data.frame(distribution = "t (df=30)", t_dist_df30),
  data.frame(distribution = "t (df=120)", t_dist_df120)
)

# Plot the distributions with a bold and thick line for the normal distribution
ggplot(data = combined_data, 
       mapping = aes(x = x, 
                     y = y, 
                     color = distribution)
       ) +
  geom_line(size = 1.5, 
            linetype = ifelse(test = combined_data$distribution == "Normal", 
                              yes = "solid",
                              no =  "dashed"
                              )
            ) +
  labs(title = "Normal and Student's t-Distributions",
       x = "Value",
       y = "Density") +
  theme_minimal()

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

Discovery based on the graph:

• With a large number of degrees of freedom, a t-distribution resembles a normal distribution.

Part II

set.seed(123)  # Set seed for reproducibility

# Set parameters for the normal distribution 
mu      <-  108
sigma <-  7.2
data_values <- rnorm(n = 1000,   mean = mu,  sd = sigma   ) 

# turn random data into a dataframe 
data <- data.frame(data_values)

# Using ggplot to graph the T-distribution 
ggplot(data = data, aes(x = data_values)) +
  geom_histogram(binwidth = 5, fill = "orange", color = "black")+
  labs(title = "OrginalNormal Distribution Histogram Plot",
       x = "Value",
       y = "Frenquency"
       ) +
  theme_classic()

# Z-scale the data
z_data <- scale(data)

# Plot the Z-scaled distribution
ggplot(data.frame(x = z_data), aes(x = data_values)) +
  geom_histogram(binwidth = 0.2, fill = "lightgreen", color = "black", alpha = 0.9) +
  labs(title = "Z-scaled Distribution",
       x = "Z-score",
       y = "Frequency") +
  theme_classic()

Part III P-value

• What is P-value?

  • It tells us how likely the data you have observed is to have occur-ed under the null hypothesis

    • High P-value: highly likely that the null-hypothesis is true

    • Low P-value: The null hypothesis is likely untrue

      • A low P-value suggests that your sample provides enough evidence that you can reject the null hypothesis.

  • Normally, ⍺, the significance value is predetermined before the testing and usually is either 5%, or 1%. If P-value is smaller than the pre-determined ⍺ value, then we can reject the null hypothesis.