Demonstration of t-Test in Statistics Using R

 

Experiment

Demonstration of t-Test in Statistics Using R

1. Experiment Title

Hypothesis Testing Using t-Test in R

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2. Aim

To study and perform a t-test using R to determine whether the mean of a sample differs significantly from a specified population mean.

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3. Objectives

After completing this experiment, students should be able to:

1. Understand the concept of hypothesis testing.
2. Formulate the null and alternative hypotheses.
3. Perform a one-sample t-test.
4. Calculate the t-statistic manually using the appropriate formula.
5. Perform the t-test using R.
6. Interpret the p-value and make a statistical decision.
7. Draw a conclusion based on the test result.

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4. Theory

4.1 What is a t-Test?

A t-test is a parametric statistical test used to determine whether there is a statistically significant difference between means.

The t-test is commonly used when:

• The population standard deviation is unknown.
• The sample size is relatively small.
• The data are quantitative.
• The observations are independent.
• The data are approximately normally distributed.

There are three commonly used types of t-tests:

1. One-sample t-test – compares a sample mean with a known or hypothesized population mean.
2. Independent two-sample t-test – compares the means of two independent groups.
3. Paired t-test – compares two related measurements.

In this experiment, we demonstrate the one-sample t-test.

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5. Problem Statement

A university claims that the average examination score of its students is 70 marks.

A sample of 10 students produced the following marks:

72, 68, 75, 71, 69, 74, 73, 70, 76, 72

Using a significance level of:

$$\alpha = 0.05$$

determine whether the sample provides sufficient evidence to conclude that the average examination score differs from 70 marks.

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6. Hypothesis Formulation

The university claims that the population mean is 70.

Null Hypothesis

$$H_0:\mu=70$$

There is no significant difference between the population mean and 70.

Alternative Hypothesis

$$H_1:\mu\neq70$$

There is a significant difference between the population mean and 70.

Since we are checking whether the mean is different from 70, this is a two-tailed test.

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7. Manual Calculation

Step 1: Calculate the Sample Mean

The sample data are:

$$72,68,75,71,69,74,73,70,76,72$$

The sample mean is:

$$\bar{x}=\frac{\sum x_i}{n}$$

The sum of the observations is:

$$72+68+75+71+69+74+73+70+76+72=720$$

The number of observations is:

$$n=10$$

Therefore:

$$\bar{x}=\frac{720}{10}=72$$

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Step 2: Calculate the Sample Standard Deviation

The sample standard deviation is:

$$s=\sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}$$

For the given observations:

$$s\approx2.582$$

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Step 3: Calculate the t-Statistic

The formula for a one-sample t-test is:

$$t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

where:

• $\bar{x}$ = sample mean
• $\mu_0$ = hypothesized population mean
• $s$ = sample standard deviation
• $n$ = sample size

Substituting the values:

$$t=\frac{72-70}{2.582/\sqrt{10}}$$ $$t\approx2.45$$

The degrees of freedom are:

$$df=n-1=10-1=9$$

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8. R Program

# One-Sample t-Test in R

# Sample examination marks
marks <- c(72, 68, 75, 71, 69,
           74, 73, 70, 76, 72)

# Hypothesized population mean
mu <- 70

# Significance level
alpha <- 0.05


# Display sample statistics

cat("Sample Size =", length(marks), "\n")
cat("Sample Mean =", mean(marks), "\n")
cat("Sample Standard Deviation =", sd(marks), "\n")


# Perform one-sample t-test

result <- t.test(marks,
                 mu = mu,
                 alternative = "two.sided",
                 conf.level = 0.95)


# Display the result

print(result)


# Decision based on p-value

if (result$p.value < alpha) {
  
  cat("\nDecision: Reject the Null Hypothesis\n")
  cat("Conclusion: The mean is significantly different from", mu)
  
} else {
  
  cat("\nDecision: Fail to Reject the Null Hypothesis\n")
  cat("Conclusion: There is not enough evidence to conclude that the mean differs from", mu)
}

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9. Explanation of Important R Function

The main function used is:

t.test()

For this experiment:

t.test(marks, mu = 70)

The function performs a one-sample t-test.

Important Arguments

Argument	Meaning
marks	Sample data
mu = 70	Hypothesized population mean
alternative = "two.sided"	Tests whether the mean is different
conf.level = 0.95	Uses a 95% confidence level

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10.Output

The output will contain information similar to:

Sample Size = 10 
Sample Mean = 72 
Sample Standard Deviation = 2.581989 

	One Sample t-test

data:  marks
t = 2.4495, df = 9, p-value = 0.03679
alternative hypothesis: true mean is not equal to 70
95 percent confidence interval:
 70.15296 73.84704
sample estimates:
mean of x 
       72 


Decision: Reject the Null Hypothesis
Conclusion: The mean is significantly different from 70

The exact p-value and confidence interval can be obtained by running the program.

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11. Interpretation of the Output

The important values obtained from a t-test are:

1. t-value

The t-value indicates how far the sample mean is from the hypothesized mean relative to the variation in the data.

2. Degrees of Freedom

For a one-sample t-test:

$$df=n-1$$

3. p-value

The p-value is used to make the statistical decision.

At:

$$\alpha=0.05$$

• If p-value < 0.05, reject $H_0$.
• If p-value ≥ 0.05, fail to reject $H_0$.

4. Confidence Interval

The confidence interval gives a range of plausible values for the population mean.

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12. Result

Thus, a one-sample t-test was performed using R to test whether the mean examination score differs significantly from the hypothesized population mean of 70. The sample mean, t-statistic, degrees of freedom, p-value, and confidence interval were obtained and used to make a statistical conclusion regarding the null hypothesis.