Understanding One-Sample Hypothesis Tests

One-sample hypothesis tests are statistical methods used to determine if a sample mean is significantly different from a known or hypothesized population mean. These tests are essential in various fields, from health sciences to market research, as they help researchers make informed decisions based on sample data. Below are three diverse examples that illustrate how one-sample hypothesis tests can be applied in real-world scenarios.

Example 1: Testing Average Blood Pressure Levels

In a health study, a researcher wants to determine if the average blood pressure of a specific population is significantly different from the national average of 120 mmHg. The researcher collects a sample of 30 individuals from this population and measures their blood pressure.

• Sample Data: Blood pressure readings (in mmHg) from the sample are: 125, 118, 132, 121, 117, 124, 128, 115, 130, 127, 123, 126, 119, 122, 129, 131, 114, 136, 133, 137, 115, 120, 135, 138, 140, 116, 139, 141, 143, 144.
• Hypothesis Statement:
  • Null Hypothesis (H0): The average blood pressure of the population is 120 mmHg.
  • Alternative Hypothesis (H1): The average blood pressure of the population is not 120 mmHg.
• Analysis: Calculate the sample mean and standard deviation, then perform a t-test.
• Conclusion: If the p-value is less than the significance level (typically 0.05), reject the null hypothesis and conclude that the average blood pressure of the population is significantly different from 120 mmHg.

Notes

This example highlights the application of one-sample hypothesis testing in public health. Variations could include testing against different population means or using different sample sizes.

Example 2: Evaluating Average Customer Satisfaction Scores

A retail chain conducts a customer satisfaction survey and wants to determine if the average satisfaction score of their customers has changed from the historical average of 4.0 (on a scale of 1 to 5).

• Sample Data: Customer satisfaction scores from a recent survey of 50 customers: 4.2, 3.9, 4.5, 4.0, 4.1, 4.3, 3.8, 4.4, 4.0, 4.6, 4.2, 3.7, 4.1, 4.0, 4.5, 4.3, 4.1, 4.0, 4.4, 3.9, 4.2, 4.0, 4.3, 4.1, 4.5, 4.2, 4.0, 3.8, 4.1, 4.4, 4.0, 4.3, 4.2, 3.9, 4.5, 4.0, 4.2, 4.1, 3.8, 4.4, 4.5, 4.0, 3.9, 4.3, 4.1, 4.0, 4.2, 4.3, 4.0, 4.1.
• Hypothesis Statement:
  • Null Hypothesis (H0): The average customer satisfaction score is 4.0.
  • Alternative Hypothesis (H1): The average customer satisfaction score is not 4.0.
• Analysis: Calculate the sample mean and standard deviation, then perform a one-sample t-test to compare the sample mean to the historical average.
• Conclusion: If the p-value is below the significance level, reject the null hypothesis, indicating a significant change in customer satisfaction.

Notes

This example demonstrates hypothesis testing in the context of customer feedback. Variations could involve testing against different historical averages or using qualitative data.

Example 3: Assessing Average Daily Study Hours

An educational researcher wants to assess whether high school students in a particular district study on average more than 2 hours per day, which is the established guideline.

• Sample Data: Daily study hours recorded from a sample of 40 students: 2.5, 1.8, 2.0, 2.3, 1.9, 2.2, 3.0, 2.1, 2.4, 1.6, 2.8, 2.7, 1.5, 2.3, 2.0, 2.2, 2.5, 2.4, 1.9, 2.1, 2.3, 2.6, 1.8, 2.9, 2.0, 1.7, 2.4, 2.1, 2.3, 2.5, 1.9, 2.2, 2.6, 3.1, 1.8, 2.4, 2.2, 2.0, 2.5, 2.1, 1.9.
• Hypothesis Statement:
  • Null Hypothesis (H0): The average daily study hours is 2 hours.
  • Alternative Hypothesis (H1): The average daily study hours is greater than 2 hours.
• Analysis: Compute the sample mean and standard deviation, then perform a one-sample t-test comparing the sample mean to 2 hours.
• Conclusion: If the resulting p-value is less than 0.05, reject the null hypothesis, suggesting that students study significantly more than 2 hours daily.

Notes

This example illustrates the application of hypothesis testing in education research. Variations could include testing other time metrics or different educational settings.