Statistical Hypothesis Testing and Markov Chain Problem Solutions

Introduction to Statistical Methods and Examples

Initial Setup and Data Visualization

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Key Concepts in Statistical Hypothesis Testing

Statistical Hypothesis

To reach decisions about populations based on sample information, we make certain assumptions about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses.

Null Hypothesis (H₀) and Alternative Hypothesis (H₁)

The hypothesis formulated for the purpose of its rejection, under the assumption that it is true, is called the Null Hypothesis, denoted by H₀. The hypothesis complementary to the null hypothesis is called the Alternative Hypothesis, denoted by H₁.

Test of Significance

The process that helps us decide about the acceptance or rejection of a hypothesis is called the Test of Significance.

Significance Levels (α)

The level of significance (α) is a measure of statistical significance. It defines the probability threshold used to decide whether the null hypothesis (H₀) is accepted or rejected. It helps identify if the result is statistically significant enough to reject H₀.

Critical Region (Rejection Region)

A critical region, also known as the rejection region, is a set of values for the test statistic for which the null hypothesis is rejected. If the observed test statistic falls within the critical region, we reject the null hypothesis and accept the alternative hypothesis.

Problem 1: Comparing Smoking Habits Between Two Cities

Testing Difference in Proportions at 5% Significance Level

In a sample of 600 men from City A, 450 are found to be smokers. In another sample of 900 men from City B, 450 are smokers. Does this indicate that the cities are significantly different with respect to the habit of smoking among men? Test at the 5% significance level.

Problem 2: Analyzing Joint Distribution of Random Variables X and Y

The joint distribution of two random variables X and Y are as follows:

Problem 3: Statistical Analysis and Calculations

Problem 4: Further Statistical Calculation

Problem 5: Determining the Fixed Probability Vector (Markov Chain)

Since the given matrix P is of order 3x3, the required fixed probability vector Q must also be of order 3x3. Let Q = [x y z].

For every x ≥ 0, y ≥ 0, z ≥ 0, we must have x + y + z = 1. Also, the fixed vector satisfies the equation: QP = Q.

Problem 6: Markov Chain Analysis of Ball Throwing

Three boys A, B, and C are throwing a ball to each other. The rules are:

• A always throws the ball to B.
• B always throws the ball to C.
• C is just as likely to throw the ball to B as to A.

If C was the first person to throw the ball, find the probabilities that after three throws:

1. A has the ball
2. B has the ball
3. C has the ball

Given the transition probability matrix (TPM) of the Markov chain:

Problem 7: Testing for Die Bias (Large Sample Test)

A die is thrown 9000 times, and a throw of 3 or 4 was observed 3240 times. Show that the die cannot be regarded as an unbiased one.