Please load these libraries

library(ggplot2)
library(knitr)
library(tinytex)
library(plotly)

Attaching package: 'plotly'

The following object is masked from 'package:ggplot2':

    last_plot

The following object is masked from 'package:stats':

    filter

The following object is masked from 'package:graphics':

    layout

What is hypothesis testing?

• A statistical method used to assess the validity of a claim (hypothesis) about a population parameter.
• It involves collecting data, analyzing it, and drawing conclusions about the population based on the sample.

Load the following Data

  Tree  age circumference
1    1  118            30
2    1  484            58
3    1  664            87
4    1 1004           115
5    1 1231           120
6    1 1372           142

Describes the dataset: circumference (circumference of orange trees) and age (age of trees)

1. State the null and alternative hypotheses.
   
2. Set the significance level (α).
   
3. Choose the appropriate statistical test.
   
4. Collect data from a random sample.
   
5. Calculate the test statistic.
   
6. Determine the p-value (probability of observing the data or more extreme, assuming H₀ is true).
   
7. Make a decision based on the p-value.
   

• Reject H₀ if p-value < α (evidence against the null hypothesis).
  
• Fail to reject H₀ if p-value ≥ α (not enough evidence to reject the null hypothesis).
  

-Null hypothesis (H₀): There is no relationship between age and orange tree circumference.
-Alternative hypothesis (H₁): There is a positive relationship between age and orange tree circumference.
-Significance level (α) = 0.05

ggplot(Orange, aes(x = age, y = circumference)) +
  geom_point(color="purple",shape=22) +
  geom_smooth(method = "lm", se = FALSE,color="red") +
  labs(title = "Orange Tree Circumference by Age", x = "Age (years)", y = "Circumference (mm)")

`geom_smooth()` using formula = 'y ~ x'

Hypothesis: -H₀: The slope of the regression line is zero (no relationship between age and circumference).
-H₁: The slope of the regression line is not zero (there is a relationship between age and circumference).
Statistical Test: Linear regression
Formula for t-statistic:
t=β^1/SE(β1)
Where:
-β^1 is the estimated slope coefficient
-SE(β^1) is the standard error of the slope coefficient

Perform the linear regression and calculate the p-value associated with the slope coefficient.
Interpret the results:
-If p-value < α, reject H₀ (there is a significant relationship between age and circumference).
-If p-value ≥ α, fail to reject H₀ (no significant relationship between age and circumference).

Based on the graph, we can conclude that the smaller circumferences has a high frequency which means there was a lot of new growth.