The 95% confidence interval represents the range of values which are likely to contain true population parameter such as population mean with a 95% confidence level. Typically, the 95% confidence interval for a large sample is calculated as: Where, x̄ is a sample mean, σ is the population standard deviation, n is the sample size, and 1.96 is a critical value for 95% confidence. The part of the equation in bracket is also called as standard error of the mean.

In Python, you can use the chi2_contingency function from the scipy package to perform a proportion test on two groups. chi2_contingency function calculates the chi-square test on a contingency table to compare the proportions between the groups. However, the chi2_contingency function does not provide built-in methods for reporting confidence intervals for differences in two proportions. But you can calculate the Wald confidence interval for differences in two Proportions from a 2x2 contingency table.

Proportion test is used for comparing the proportions (e.g. number of successes) in two or more groups to determine if there are significant differences between these groups. In Python, you can use the chi2_contingency function from the scipy package to perform a proportion test similar to the prop.test function in R. chi2_contingency function performs a chi-squared test of independence (similar to the prop.test function in R) based on proportions provided in the contingency table.

Both pandas apply and NumPy vectorize functions are useful in manipulating the pandas DataFrame, but these functions can have specific uses and performance characteristics. pandas apply function can be used to apply built-in or custom functions along an axis of the DataFrame. pandas apply function is very flexible and can be used for applying complex manipulations such as calculations with conditional logic on the pandas DataFrame. However, pandas apply function is limited by the performance issue.

pandas apply function is widely used to apply custom functions on rows and columns of the DataFrame. The apply function is like a loop function which iterates through each element of rows or columns based on a given axis, and applies the given function. The basic syntax of the apply function is: pd.DataFrame.apply(function, axis=0) The axis parameter defines the axis where the function is applied. The axis=0 means the function is applied on each column.

Z-score (also known as standard score) is a statistical measure that calculates how many standard deviations a data point from the mean of the data distribution. In pandas DataFrame, you can calculate the Z-score for one or all columns using the zscore function from the SciPy Python package or by manual method. The following example demonstrates how to calculate the Z-score for all numeric columns in a pandas DataFrame. Using zscore function Create a random pandas DataFrame,