Statistical concepts are commonly asked in data science and machine learning interviews. In this tutorial, we’ll delve into the top 12 interview questions related to these concepts and provide guidance on how to effectively answer them. The Python code for simulating the statistical concepts will also be provided where is applicable.

**Resources for this post:**

- Video tutorial for this post on YouTube
- Click here for the Colab notebook
- More video tutorials on Data Science Interview Questions and Statistics
- More blog posts on Data Science Interview Questions and Statistics

Let’s get started!

### Question 1: What is the central limit theorem?

The Central Limit Theorem states that the sampling distribution of the mean of a large number of independent, identically distributed random samples will approach a normal distribution, regardless of the original population’s distribution.

Here’s a sample Python code to demonstrate the Central Limit Theorem using the `numpy`

and `matplotlib`

libraries:

import numpy as np

import matplotlib.pyplot as plt

# Define population parameters

population_mean = 50

population_std = 10

population_size = 1000

# Generate a population with a non-normal distribution

population = np.random.gamma(population_mean, population_std, population_size)

# Define sample parameters

sample_size = 50

num_samples = 1000

# Draw multiple samples and calculate their means

sample_means = [np.mean(np.random.choice(population, sample_size)) for _ in range(num_samples)]

# Plot the histogram of sample means

plt.hist(sample_means, bins=20, density=True)

plt.xlabel('Sample Mean')

plt.ylabel('Frequency')

plt.title('Central Limit Theorem Demonstration')

plt.show()

This code creates a non-normal population, takes multiple samples from the population, calculates the means of these samples, and then plots the distribution of the sample means, illustrating the Central Limit Theorem in action.

### Question 2: What is the law of large numbers?

The Law of Large Numbers states that as the number of trials or observations in a random experiment increases, the sample average converges to the expected value, or true population mean.

Here’s a sample Python code to demonstrate the Law of Large Numbers using the `numpy`

and `matplotlib`

libraries:

import numpy as np

import matplotlib.pyplot as plt

# Define population parameters

population_mean = 50

population_std = 10

population_size = 1000

# Generate a population with a normal distribution

population = np.random.normal(population_mean, population_std, population_size)

# Define the number of trials

num_trials = [10, 50, 100, 500, 1000, 5000, 10000]

# Calculate the sample averages for different numbers of trials

sample_averages = [np.mean(np.random.choice(population, n_trials)) for n_trials in num_trials]

# Plot the sample averages vs. the number of trials

plt.plot(num_trials, sample_averages, 'o-', label='Sample Average')

plt.axhline(population_mean, color='r', linestyle='--', label='Population Mean')

plt.xlabel('Number of Trials')

plt.ylabel('Sample Average')

plt.title('Law of Large Numbers Demonstration')

plt.legend()

plt.show()

This code creates a normal population, takes multiple samples from the population with different numbers of trials, calculates the sample averages, and then plots the sample averages versus the number of trials, illustrating the Law of Large Numbers.

### Question 3: What is a p-value?

P-value measures the probability of obtaining results that are at least as extreme as the observed results if the null hypothesis is true. P-value ranges from 0 to 1.

- A high p-value means that there is a high chance of obtaining the results as extreme as observed if the null hypothesis is true, so we fail to reject the null hypothesis.
- A low p-value means that there is a low chance of obtaining the results as extreme as observed if the null hypothesis is true, so we reject the null hypothesis.
- The typical threshold for p-value is 0.05, meaning that if the probability of obtaining the results as extreme as observed given the true null hypothesis is less than 0.05, we reject the null hypothesis.

Here’s a sample Python code to calculate the p-value using a one-sample t-test with the SciPy library. This code tests the hypothesis that the sample mean is equal to a given value, using a significance level of 0.05:

import numpy as np

from scipy import stats

# Define the sample data

sample_data = [48, 52, 45, 50, 51, 47, 49, 53, 46, 55]

# Define the null hypothesis mean

hypothesized_mean = 50

# Perform a one-sample t-test

t_stat, p_value = stats.ttest_1samp(sample_data, hypothesized_mean)

# Define the significance level

alpha = 0.05

# Check if the p-value is less than the significance level

if p_value < alpha:

print(f"Reject the null hypothesis (p-value: {p_value:.4f})")

else:

print(f"Fail to reject the null hypothesis (p-value: {p_value:.4f})")

Output:

Fail to reject the null hypothesis (p-value: 0.7022)

This code uses the `ttest_1samp`

function from the SciPy library to perform a one-sample t-test on the given sample data and calculate the p-value. It then compares the p-value with the significance level (alpha) to determine whether to reject or fail to reject the null hypothesis.

### Question 4: What is the standard deviation?

Here’s a simple Python code to calculate the standard deviation of a dataset using the `numpy`

library:

import numpy as np

# Define the dataset

data = [48, 52, 45, 50, 51, 47, 49, 53, 46, 55]

# Calculate the standard deviation

standard_deviation = np.std(data, ddof=1)

print(f"Standard deviation: {standard_deviation:.2f}")

Output:

Standard deviation: 3.20

In this code, we use the std function from the `numpy`

library to calculate the standard deviation of the given dataset. The `ddof`

parameter is set to 1 to compute the sample standard deviation; `ddof`

is Delta Degrees of Freedom for population standard deviation.

### Question 5: What is a confidence interval?

- Confidence interval is used when we would like to know a population parameter but only have access to samples. Since the true population value is unknown, we do not know if the sample value is greater than, less than, or equal to the population parameter.
- A 95% confidence interval can be interpreted as

- We are 95% confident that the true population parameter is within CI lower bound and upper bound.
- If we repeat the sampling 100 times, 95 times out 100 the true population value is within the confidence interval.

- Suppose we want to determine the mean height of all adults in the United States. We can take multiple samples from the adult population, compute the sample mean, and construct a confidence interval. A 95% confidence interval indicates that if we were to repeat this process 100 times, we would expect the true average height of the US adult population to fall within the interval 95 out of 100 times.
- What affects the size of the confidence interval?

- Variation: A population with low variation leads to samples with low variation and a narrower confidence interval.
- Sample size: As the sample size increases, the CI becomes smaller, assuming other factors remain constant.

- How to calculate the confidence interval?

Here’s a sample Python code to calculate the confidence interval for a given sample using the t-distribution from the SciPy library:

import numpy as np

from scipy import stats

# Define the sample data

sample_data = [48, 52, 45, 50, 51, 47, 49, 53, 46, 55]

# Calculate the sample mean and standard error

sample_mean = np.mean(sample_data)

sample_std = np.std(sample_data, ddof=1) # ddof is Delta Degrees of Freedom

sample_size = len(sample_data)

standard_error = sample_std / np.sqrt(sample_size)

# Define the desired confidence level

confidence_level = 0.95

# Calculate the degrees of freedom

degrees_of_freedom = sample_size - 1

# Calculate the t-distribution critical value

t_critical = stats.t.ppf((1 + confidence_level) / 2, degrees_of_freedom)

# Calculate the confidence interval

margin_of_error = t_critical * standard_error

confidence_interval = (sample_mean - margin_of_error, sample_mean + margin_of_error)

print(f"{confidence_level * 100}% confidence interval: {confidence_interval}")

Output:

95.0% confidence interval: (47.30787918518414, 51.89212081481586)

This code calculates the sample mean, standard error, degrees of freedom, and t-distribution critical value. It then computes the confidence interval for the given sample data using the calculated values.

### Question 6: What are Type I and Type II errors?

- Type I error is the same as the significance level or false positive rate. It is when the null hypothesis is rejected when it is true.
- Type II error is the same as false negative rate. It is when the null hypothesis is not rejected when it is false. The values of type II error equals 1 minus power in hypothesis testing. Power is the probability of detecting the difference if it exists. It represents how confident we are about rejecting the null hypothesis

### Question 7: What’s the difference between a covariance and a correlation?

**Covariance**measures the degree to which two variables change together, indicating the direction of the relationship. A positive covariance means that as one variable increases, the other tends to increase as well, while a negative covariance indicates that as one variable increases, the other tends to decrease. However, covariance does not provide information about the strength of the relationship, and its value depends on the scale of the variables, making it difficult to compare covariances between different datasets.**Correlation**is a standardized version of covariance that measures both the direction and strength of the linear relationship between two variables. The correlation coefficient, typically denoted by`r`

or`ρ`

, ranges from -1 to 1. A value of -1 indicates a perfect negative linear relationship, 1 indicates a perfect positive linear relationship, and 0 indicates no linear relationship. Since correlation is unitless and standardized, it allows for easy comparison between different datasets and variables.

### Question 8: What’s the difference between a Z-test and a T-test?

**Z-test**is used when the population standard deviation is known, and the sample size is large (typically, n > 30). The Z-test assumes that the underlying population is normally distributed, or the sample size is large enough for the Central Limit Theorem to hold. In a Z-test, the test statistic follows the standard normal distribution (Z-distribution), which has a mean of 0 and a standard deviation of 1.**t-test**is used when the population standard deviation is unknown, and the sample size is small (typically, n <= 30). The t-test assumes that the sample is drawn from a population with a normal distribution. In a t-test, the test statistic follows a t-distribution, which is similar to the standard normal distribution but has thicker tails. The t-distribution is characterized by the degrees of freedom, which depend on the sample size. As the sample size increases, the t-distribution approaches the standard normal distribution.

### Question 9: What is the expected value of a binomial distribution?

- A binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is used to describe situations where there are only two possible outcomes in each trial, often referred to as “success” and “failure.”

### Question 10: What’s the impact of removing the data below the mean of a normally distributed dataset?

- Inflated Type 1 error rate: Removing values below the mean or any other threshold alters the distribution’s characteristics and may violate the assumptions of the statistical test being used. If the test is not robust to these changes, the Type 1 error rate (the probability of rejecting a true null hypothesis) could be inflated.
- Non-normality: If the original sampling distribution was approximately normal, removing values below the mean will lead to a non-normal distribution, which might not be suitable for parametric tests like t-tests or Z-tests that rely on the normality assumption.
- Reduced sample size: Filtering out values will reduce the sample size, which can decrease the statistical power of the test. This means you might be more likely to commit a Type 2 error (failing to reject a false null hypothesis) while potentially increasing the Type 1 error rate.
- Biased parameter estimates: By removing values below the mean, you introduce bias into the sample mean and variance estimates, which can affect the test statistic and p-value calculations, leading to incorrect conclusions about the null hypothesis.
- Change in effect size: If the goal is to estimate the effect size or the magnitude of a relationship between variables, removing values below the mean will likely change the estimated effect size and might lead to erroneous conclusions.

### Question 11: What is the standard error of the mean?

- The standard error quantifies the precision of a sample statistic as an estimate of the population parameter.

- A smaller standard error indicates that the sample statistic is a more precise estimate of the population parameter, while a larger standard error suggests more variability between samples. The standard error is used in various statistical tests, such as hypothesis testing and confidence interval calculations, to assess the reliability of the sample estimates and make inferences about the population.

### Question 12: What is the Margin of Error?

- The margin of error represents the range within which the true population value is likely to fall, given the observed sample statistic and a specified level of confidence.
- The margin of error is typically calculated as the product of the critical value (from a chosen probability distribution, such as the Z-distribution for large samples or the t-distribution for small samples) and the standard error of the sample statistic. For a sample mean, the margin of error can be calculated as: 𝑀𝑎𝑟𝑔𝑖𝑛𝑜𝑓𝐸𝑟𝑟𝑜𝑟=𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙𝑉𝑎𝑙𝑢𝑒∗𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝐸𝑟𝑟𝑜𝑟
- where the critical value depends on the desired level of confidence (e.g., 1.96 for a 95% confidence level in the case of a Z-distribution).

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