Statistics: Central Limit Theorem and Hypothesis Testing

Name: Statistics: Central Limit Theorem and Hypothesis Testing
Rating: 4.6 (92 reviews)

Get a thorough understanding of the most important concepts in Statistics - Central Limit Theorem and Hypothesis Testing

Created byShubham Kalra

Last updated 9/2020

English

What you'll learn

Understand Normal & Standard Normal Distribution and feel much more confident in solving the questions
Build a good intuitive understanding of Central Limit Theorem - One of the most important concepts in Statistics
Understand the basics and essence of Hypothesis Testing
Solve exam style questions in a step by step manner with much more confidence

Course content

4 sections • 60 lectures • 6h 58m total length

Introduction3:38
Explore continuous random variables and probability density functions that assign probabilities to ranges rather than individual values. Compare discrete and continuous distributions and note that single-value probabilities are virtually zero.
Introduction (Written Notes)0:06
Uniform Distribution9:39
Examine the uniform distribution as a continuous, rectangular model, derive the density f(x)=1/(b−a) on [a,b], and compute equal interval probabilities like 100–120 minutes.
Uniform Distribution (Written Notes)0:07
Area as a measure of probability5:40
Compute probabilities for continuous random variables by measuring the area under the graph between values. Use this approach on normal distributions, noting that a single point has zero probability.
Normal Distribution1:24
Examine the normal distribution’s bell-shaped curve, centered at the mean, with spread set by the standard deviation sigma, while avoiding direct use of its probability density function.
Characteristics of Normal Distribution9:17
Explore the normal distribution by its symmetry, mean–median–mode equality, and the role of mean and standard deviation in location and spread; 68.3%, 95.4%, 99.7% fall within 1–3 sd.
Standard Normal Distributions2:28
Learn about the standard normal distribution with mean zero and standard deviation one, its density function, and using z-tables to calculate probabilities as areas under the curve.
How to calculate probability using Cumulative Probability table6:13
Learn to compute probabilities using the standard normal cumulative table by locating z in the table and reading the area under the curve.
2 important rules to remember2:44
Explore two core rules of the standard normal distribution: symmetry about zero and the complement rule, using probabilities and area under the curve to relate opposite regions.
3 types of probability calculation9:10
Explore the three types of probability for the standard normal: P(Z ≤ x), P(Z ≥ x), and P(a ≤ Z ≤ b), using symmetry and the cumulative distribution table.
How to compute z values when probability value is given3:25
Compute z values from given probabilities using the cumulative normal table by shading the left area under the standard normal curve; recognize two question types.
A different type of table to compute probability6:46
Learn a different probability table that gives the area between 0 and a value under the standard normal distribution, and read it like the cumulative table.
How to calculate probability for any normal distribution10:42
Convert any normal distribution to the standard normal with z = (x − μ)/σ, then compute probabilities from the standard normal table.
Application of Normal Distribution16:28
Apply the normal distribution to tire mileage data by computing probabilities and threshold values using z-scores, mean, and standard deviation, with examples of exceeding 40,000 miles and a 10% guarantee.
Revisiting the application15:32
Learn to find z-values from a normal distribution using interpolation on cumulative tables, including handling unseen probabilities and applying symmetry for negative z-scores.
Normal Distribution : Some Real Life Examples1:24
Normal Distribution using Microsoft Excel7:52
Learn to compute normal distribution probabilities in Microsoft Excel using norm.dist and not functions, with mean 36500 and standard deviation 5000, to find the probability x exceeds 40000.
Practice Question 111:19
Solve a normal distribution question by converting to standard normal z-scores, noting that P(X=105)=0, calculating P(X<105)=0.5793, and identifying the most extreme 0.1 percent values.
Practice Question 25:46
This question uses a normal distribution for speed (mean 46.8, sd 1.75) and computes P(X ≤ 50) ≈ 0.9664, P(X ≥ 48) ≈ 0.245, and P(|X−μ| ≤ 1.5σ) ≈ 0.868?
Practice Question 317:41
Analyze a normal distribution with mean 8.8 and sd 2.8 to compute P(X≥10), P(X>20), and P(5<X<10); find c for 0.98 interval; probability at least one of four trees exceeds 10.
Practice Question 47:49
Compare two cog-diameter machines by normal distributions, computing P(2.9< X <3.1) for X~N(3,0.1^2) and P(2.9< V <3.1) for V~N(3.0,0.02^2); conclude machine two is more likely to produce an acceptable cog.
Practice Question 57:11
Use a normal model with mean 43 and sd 4.5, applying z-scores. Find P(X ≤ 40) ≈ 0.251 and P(X > 60) ≈ 0, and the 25th percentile ≈ 39.99.
Practice Question 65:38
Determine the damage probability for a parachute with X ~ N(200,30); P(X ≤ 100) ≈ 0.0004, then compute at least one damage among five parachutes as ≈ 0.002.
Practice Question 74:36
Bearings follow a normal distribution with mean 0.499 and standard deviation 0.002; within the 0.496–0.504 range, the acceptance probability is 0.927, making 7.3% not acceptable.
Practice Question 812:53
Apply normal distribution with mean 70 and sd 3 to find the probability hardness lies between 67 and 75 using standard normal tables, then apply binomial concepts for 10 specimens.
Practice Question 96:22
Use the normal approximation to a binomial with n=2000 and p=0.03; mu=30, sigma≈5.39, to find P(X≥40) ≈ 0.039. The probability of at most 50 detectors is essentially 1.
Share your experience0:29

Introduction to the Section3:03
The Sampling Problem4:33
Explore the sampling problem: compare population data to a random sample of managers to estimate the mean salary and population parameters, and learn how to select a good random sample.
Simple Random Sample - Finite Population9:59
Explore simple random sampling from a finite population, ensuring every manager has an equal chance. See two practical methods, hats and Excel tools, to draw a 30-manager random sample.
Simple Random Sample - Infinite Population3:20
Explore simple random sampling for infinite populations, emphasize independent selection and avoiding selection bias, illustrated by a McDonald's coupon example.
How to calculate the point estimators of population parameters?5:52
Estimate population parameters by computing the sample mean and sample standard deviation from a sample, as point estimators for the population mean and standard deviation.
Sampling Distribution8:59
Observe how sample means from random samples form a sampling distribution with a mean and standard deviation. The distribution appears near normal, illustrating the central limit theorem.
Properties of Sampling Distribution6:38
Discover that the expected value of x-bar equals the population mean and how finite population correction shapes the standard deviation in the sampling distribution.
Central Limit Theorem Explained5:58
Demonstrates that the sampling distribution of the sample mean is approximately normal for large enough samples, regardless of population shape. Illustrates how larger sample sizes improve normality, focusing on x-bar.
Why large samples are considered to be better predictors of population parameter12:48
Demonstrate how larger samples provide closer estimates of the population mean by applying the central limit theorem, analyzing sampling distributions and the shrinking standard error.
Share your experience0:29

Basics of Hypothesis Testing10:53
Test hypotheses about bottle content by comparing sample means to 250 ml claim with null and alternative hypotheses and a rejection region, noting means are normal by central limit theorem.
Null and Alternate hypothesis6:09
Identify the claim and counterclaim to set up null and alternative hypotheses, using the equality form for the null (mu = 250, mu = 36) in examples.
Burden of proof0:53
Type of test and Rejection Region11:33
Identify left-tail, right-tail, and two-tailed hypothesis tests and how the rejection region guides rejecting the null. Use examples with sample means and normal distributions to illustrate.
Types of Errors2:40
Identify the two error types in hypothesis testing: type I errors and type II errors. Learn how sample information leads to rejecting or not rejecting the null and alternative hypotheses.
Type 1 Error5:22
Learn about type I error in hypothesis testing, including the null hypothesis, alpha levels, and rejection regions, with examples and the tradeoff with type II error.
Type 2 Error2:51
Explore the type 2 error in hypothesis testing, including not rejecting the null when false, with examples from soft drink companies and criminal trials, and the beta value.
Example 1 - Critical Value Method15:40
Apply the critical value method for a one-sided test under the central limit theorem, with n=30, sigma=10, alpha=0.05, and sample mean 240 to reject the mean at least 250.
Example - Z Score Method6:01
Apply the z-score method to test a mean claim and see how an observed mean of 240 against 250 leads to rejection, comparing it with the critical value approach.
Example 2 - Critical Value Method8:55
Use the critical value method to test if the mean exceeds 20 hours, with n=36, sigma=8, alpha=0.01; 22 is below the 23.10 cutoff, so fail to reject H0.
Example 3 - Critical Value Method8:49
Example 3 applies the critical-value method in a two-tailed z test under the central limit theorem, testing mu=36 months; with n=40 and sigma=3, 34 months rejects the null.
Introduction to p-value6:53
Explore the p-value method in hypothesis testing, defining p-value as the probability of observing a result at least as extreme under the null, and compare it to alpha to decide.
Example 1 : p-value method9:05
Compute the p-value for a left-tailed test of the mean against 250 ml. Reject the null and conclude the mean is less than 250 ml.
Example 2 : p-value method7:41
Apply the p-value method to test if the average study hours exceed 20, with n=36 and sigma=8; p=0.0668 exceeds 0.01, so we fail to reject the null hypothesis.
Example 3 : p-value method12:01
Apply the two-tailed p-value method in a z-test with known sigma, doubling the tail probability; illustrated by testing a 36-month mean with n=40 against observed 34.
Practice Question 111:12
This practice question uses a two-tailed t-test to assess whether the true mean diameter of ball bearings equals 0.5 inches at 0.05 significance, with df = n minus 1.
Practice Question 25:25
Use a one-sided z-test with n=45, xbar=52.7, s=4.8 to test mu>50; z=3.77 exceeds 1.645, so reject H0 and conclude mean exceeds 50.
Practice Question 314:06
Conduct a small-sample left-tailed t-test to see if the mean toothpaste remaining is under 10 percent. Use five-tube data to compute sample mean and sd and compare at alpha 0.05.
Practice Question 48:33
Test whether the population mean exceeds 25 using a right-tailed t-test with n=5, x̄=27.5, s=5.47, α=0.05; t ≈ 1.04 is less than 2.13, so fail to reject H0.
Practice Question 59:11
Apply the central limit theorem to a left-tailed z-test of the population mean daily zinc intake, testing mu < 15 with n 115, x-bar 11.3, s 6.43.
Share your experience0:29

Requirements

Basics of Statistics (Random Variable, Probability Distributions etc.)
Knowledge of MS Excel (Preferred, not necessary)

Description

This course is the key to build an excellent understanding of Inferential Statistics. It has students (from over 100 countries) and here is what some of them have to say:

"Well explained sir, every concept is clear as water and the way you explain is very easy to understand the concept. Worth buying this course" ~ K Roshnishree

"The explanations are quite intuitive and the best part is that the course includes practice problems which helps in building the concepts" ~ Swati Sahu

"The detail level coverage of the basic topics is amazing" ~ Rehana Shake

"This instructor is doing a fine job explaining the statistics concepts" ~ Frank Herrera

"Very clear examples, thank you sir!" ~ Gitartha Pathak

Course Description:

This course is designed for students who are struggling with Statistics or who are complete beginners in statistics.

How is this course structured?

Section 1 and 2: These 2 sections cover the concepts that are crucial to understand the basics of hypothesis testing - Normal Distribution, Standard Normal Distribution, Sampling, Sampling Distribution and Central Limit Theorem. (Before you start hypothesis testing, make sure you are absolutely clear with these concepts)
Section 3: This section caters to the basics of hypothesis testing with three methods - Critical Value Method, Z-Score Method and p-value method.

My approach is hands on: Concepts, examples and solved problems addressing all the concepts covered in the lectures.

Note : Only Hypothesis Testing in Case of Single Population Mean is covered

Who this course is for:

Students who are new to statistics
Students who are struggling with statistics
Students who want a refresher of important statistics concepts in a simple and detailed manner

Statistics: Central Limit Theorem and Hypothesis Testing

What you'll learn

Explore related topics

Course content

Continuous Probability Distributions - Uniform and Normal Distribution28 lectures • 3hr 12min

Sampling and Sampling Distributions10 lectures • 1hr 2min

Hypothesis Testing21 lectures • 2hr 44min

Bonus Section1 lecture • 1min

Requirements

Description

Who this course is for: