For the following data...

1. Provide evidence as to whether this data set if Gaussian or non Gaussian.

2. Determine the normal reference range.

3. Define a cut-off. Describe how a ROC curve can be used to determine such.

95 95 81 95 98 79 88 98 84 79 98 84 81 96 102 86 90 101 83 95 94 88 97 95 84 99 98 99 79 84 85 88 101 102 91 99 98 79 99 96 88 102 102 100 101 101 101 79 89 97 90 86 105 97 99 96 95 88 89 92 93 101 100

To answer the questions, we will perform the following steps:

1. Checking for Gaussian distribution:

- Plot a histogram or a density plot of the data.
- If the distribution appears to be bell-shaped, symmetrical, and closely follows a normal distribution curve, then the data set can be considered Gaussian (or normally distributed).
- If the distribution does not appear to be bell-shaped or is skewed, then the data set can be considered non-Gaussian (or non-normally distributed).

2. Determining the normal reference range:

- Calculate the mean (average) and standard deviation of the data set.
- The normal reference range, also known as the "normal range" or "reference interval," is typically defined as ±2 standard deviations from the mean.
- Calculate the lower limit by subtracting 2 times the standard deviation from the mean.
- Calculate the upper limit by adding 2 times the standard deviation to the mean.
- The normal reference range will cover 95% of the data if it is normally distributed.

3. Defining a cut-off and using an ROC curve:

- A cut-off is a threshold value used to separate positive and negative outcomes or to classify data into two groups.
- An ROC (Receiver Operating Characteristic) curve is a graphical plot that illustrates the diagnostic ability of a binary classifier system.
- To determine a cut-off using an ROC curve, you need a binary response variable (e.g., presence or absence) and a predicted probability or a continuous score representing the likelihood of the response variable.
- Calculate the sensitivity (true positive rate) and specificity (true negative rate) for different cut-off points.
- Plot the sensitivity against 1 - specificity on an ROC curve.
- The optimal cut-off point is where the curve is closest to the top-left corner, representing high sensitivity and specificity.

Now, let's proceed with the steps for your data set:

1. Checking for Gaussian distribution:

- Plot a histogram or density plot of the data:
```
import matplotlib.pyplot as plt

data = [95, 95, 81, 95, 98, 79, 88, 98, 84, 79, 98, 84, 81, 96, 102, 86, 90, 101, 83, 95, 94, 88, 97, 95, 84, 99, 98, 99, 79, 84, 85, 88, 101, 102, 91, 99, 98, 79, 99, 96, 88, 102, 102, 100, 101, 101, 101, 79, 89, 97, 90, 86, 105, 97, 99, 96, 95, 88, 89, 92, 93, 101, 100]

plt.hist(data, bins='auto')
plt.xlabel('Values')
plt.ylabel('Frequency')
plt.title('Histogram of Data')
plt.show()
```

2. Determining the normal reference range:

- Calculate the mean and standard deviation of the data set:
```
import statistics

mean = statistics.mean(data)
std_dev = statistics.stdev(data)

print("Mean:", mean)
print("Standard Deviation:", std_dev)
```

- Calculate the lower and upper limits of the normal reference range:
```
lower_limit = mean - 2 * std_dev
upper_limit = mean + 2 * std_dev

print("Normal Reference Range (Lower Limit):", lower_limit)
print("Normal Reference Range (Upper Limit):", upper_limit)
```

3. Defining a cut-off and using an ROC curve:

- This step is not applicable since we do not have a binary response variable or predicted probabilities.

After performing these steps and analyzing the data, you should have the answers to questions 1 and 2. Remember that the interpretation of the data as Gaussian or non-Gaussian may be subjective, and additional statistical tests can be performed for a more thorough analysis if necessary.