Hypothesis Testing and Inference: Practice Problems in Statistics

Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, then using sample statistics to determine whether to reject the null hypothesis.

• Null Hypothesis (H0): The default assumption that there is no effect or no difference.

• Alternative Hypothesis (H1): The hypothesis that there is an effect or a difference.

• Test Statistic: A value calculated from sample data used to decide whether to reject H0.

• Significance Level (α): The probability of rejecting H0 when it is true (Type I error).

• Critical Value: The threshold value that the test statistic is compared to in order to make a decision.

Example: Suppose we wish to test the hypotheses H0: μ = 10 versus H1: μ ≠ 10. If the test statistic t = -2.54 and the critical value at α = 0.05 is ±1.75, we compare the test statistic to the critical value to determine the correct statistical decision.

• If |t| > critical value, reject H0.

• If |t| ≤ critical value, fail to reject H0.

Additional info: The example above is a two-tailed test for the population mean using the t-distribution.

Inference from Two Samples

Comparing Two Means

Statistical inference from two samples is used to compare the means of two populations. This is commonly done using the two-sample t-test, which determines whether the difference between sample means is statistically significant.

• Sample Mean (x̄): The average value in a sample.

• Sample Standard Deviation (s): Measures the spread of sample data.

• Sample Size (n): The number of observations in each sample.

• Test Statistic for Difference of Means: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• P-value: The probability of observing a test statistic as extreme as, or more extreme than, the observed value under H0.

Example: Given two samples with the following data:

• Sample 1: n = 25, mean = 130, s = 15

• Sample 2: n = 25, mean = 125, s = 20

To test for a difference in means, calculate the test statistic using the formula above and compare to the critical value or use the p-value for decision making.

Decision Making in Hypothesis Testing

Statistical Decisions

After calculating the test statistic, the next step is to make a decision regarding the hypotheses.

• Reject H0: If the test statistic falls in the critical region (beyond the critical value), we reject the null hypothesis.

• Fail to Reject H0: If the test statistic does not fall in the critical region, we do not have enough evidence to reject the null hypothesis.

• Significance Level (α): Common choices are 0.05 or 0.01, representing 5% or 1% risk of Type I error.

Example: If t = -2.54 and the critical value is ±1.75 at α = 0.05, since |t| > 1.75, we reject H0 and conclude there is a significant difference.

Summary Table: Hypothesis Testing Steps

Step	Description
1. State Hypotheses	Formulate H0 and H1
2. Choose Significance Level	Select α (e.g., 0.05)
3. Compute Test Statistic	Calculate t or z value from sample data
4. Determine Critical Value	Find threshold from statistical tables
5. Make Decision	Compare test statistic to critical value and decide to reject or fail to reject H0

Key Formulas

• Test Statistic for One Sample Mean: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• Test Statistic for Two Sample Means (Independent Samples): $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Additional info: These formulas assume normality and, for the two-sample test, independent samples.