Overview 

Approximating Crop Temperature Data with a Binomial Distribution 

Context: In the analysis of the crop temperature dataset, the goal is to model the number of occurrences of temperatures falling into specific bins using a Binomial distribution.

Dataset Summary:

1. Total Observations: 2200

2. Temperature Bins:

   • 0: 8.8257 - 11.5064

   • 1: 11.5064 - 14.1872

   • 2: 14.1872 - 16.8679

   • 3: 16.8679 - 19.5487

   • 4: 19.5487 - 22.2295

   • 5: 22.2295 - 24.9102

   • 6: 24.9102 - 27.5910

   • 7: 27.5910 - 30.2717

   • 8: 30.2717 - 32.9525

   • 9: 32.9525 - 35.6332

   • 10: 35.6332 - 38.3140

   • 11: 38.3140 - 40.9947

   • 12: 40.9947 - 43.6755

Experimental Probabilities for Temperature Bins: The dataset provides observed probabilities for each temperature bin.

Estimated Binomial Probabilities: Using a Binomial distribution model with n = 13 (number of bins) and probability derived from the dataset, the binomial probabilities are calculated for each bin.

Analysis 

To evaluate whether the temperature data follows a Binomial distribution, we compare the experimental probabilities of temperature occurrences in each bin with those estimated from a Binomial distribution model.

1. Define the Random Variable:

   • Random Variable X: Number of occurrences of temperatures falling into each bin.

2. Determine Parameters for Binomial Distribution:

   • Number of Trials (n): In this context, n corresponds to the number of bins, which is 13.

   • Probability of Success (p): This is the probability that a temperature falls into a particular bin. It is approximated by the proportion of temperatures observed in each bin.

3. Binomial Probabilities:

   • The Binomial distribution is used with parameters n = 13 and p = 0.258 derived from the dataset. The probabilities P(X=x) for each bin are calculated.

4. Experimental and Binomial Probabilities:

   • Experimental Probabilities: These are directly obtained from the dataset, representing the frequency of temperatures falling into each bin.

   • Binomial Probabilities: These are estimated using the Binomial model with the calculated probability p = 0.258.

   • By comparing the experimental probabilities with the binomial probabilities, we observe the following:

     • For most bins, especially those where the frequency of temperatures is higher, the binomial probabilities closely match the experimental probabilities.

     • For bins with very low frequencies (e.g., higher bins with fewer occurrences), both experimental and binomial probabilities approach zero and align closely.

   • The Binomial distribution effectively captures the general distribution pattern of temperatures across the bins, confirming that the observed data aligns well with the binomial model.

5. Insights and Observations:

   • The alignment between experimental and binomial probabilities indicates that the temperature occurrences in the dataset can be reasonably modeled using a Binomial distribution.

   • The fit is particularly good for bins with moderate frequencies, where the binomial distribution provides a robust approximation.-

Visualization

Summary 

In the analysis of the crop temperature dataset, the goal was to model temperature occurrences in predefined bins using a Binomial distribution. With 2200 observations divided into 13 bins, we compared experimental probabilities from the dataset with those estimated using the Binomial model. The analysis showed that the binomial probabilities closely matched the experimental probabilities, especially for bins with moderate frequencies. This suggests that the Binomial distribution effectively captures the temperature distribution, providing a strong approximation for modeling temperature occurrences in the dataset.