📈Preparatory Statistics Unit 12 – Intro to Hypothesis Testing: One-Sample

What's Hypothesis Testing?

• Statistical method used to make decisions or draw conclusions about a population based on sample data
• Involves formulating a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$)
• Calculates a test statistic and p-value to determine whether to reject or fail to reject the null hypothesis
• Allows researchers to assess the strength of evidence against the null hypothesis
• Helps in making data-driven decisions in various fields (psychology, medicine, business)
  • Example: Testing the effectiveness of a new drug compared to a placebo
• Requires specifying a significance level ($\alpha$) to control the Type I error rate
• Relies on the assumption that the sample is representative of the population
• Can be performed using different types of tests depending on the nature of the data and the research question

Key Terms and Concepts

• Null hypothesis ($H_0$): A statement of no effect or no difference, assumed to be true unless evidence suggests otherwise
• Alternative hypothesis ($H_a$): A statement that contradicts the null hypothesis, representing the researcher's claim
• Test statistic: A value calculated from the sample data used to make a decision about the null hypothesis
  • Examples: z-score, t-score, chi-square, F-statistic
• p-value: The probability of obtaining a test statistic as extreme as or more extreme than the observed value, assuming the null hypothesis is true
• Significance level ($\alpha$): The predetermined probability threshold for rejecting the null hypothesis, typically set at 0.05 or 0.01
• Type I error: Rejecting the null hypothesis when it is actually true (false positive)
• Type II error: Failing to reject the null hypothesis when it is actually false (false negative)
• Power: The probability of correctly rejecting a false null hypothesis
• Critical value: The value of the test statistic that separates the rejection and non-rejection regions, determined by the significance level and the distribution of the test statistic

Steps in Hypothesis Testing

1. State the null and alternative hypotheses
   • Clearly define the population parameter of interest and the hypothesized value
2. Choose the appropriate test statistic and distribution
   • Consider the type of data, sample size, and assumptions about the population
3. Specify the significance level ($\alpha$)
   • Determine the acceptable level of Type I error
4. Calculate the test statistic from the sample data
   • Use the appropriate formula based on the chosen test
5. Determine the p-value or compare the test statistic to the critical value
   • Use statistical software, tables, or calculators to find the p-value or critical value
6. Make a decision to reject or fail to reject the null hypothesis
   • If the p-value is less than $\alpha$ or the test statistic falls in the rejection region, reject $H_0$; otherwise, fail to reject $H_0$
7. Interpret the results in the context of the research question
   • Discuss the implications of the decision and any limitations of the study

Types of Hypotheses

• One-tailed (directional) hypotheses:
  • Left-tailed: The alternative hypothesis states that the population parameter is less than the hypothesized value
    • Example: $H_a: \mu < 100$
  • Right-tailed: The alternative hypothesis states that the population parameter is greater than the hypothesized value
    • Example: $H_a: \mu > 100$
• Two-tailed (non-directional) hypotheses:
  • The alternative hypothesis states that the population parameter is not equal to the hypothesized value
    • Example: $H_a: \mu \neq 100$
• Null hypothesis:
  • Always includes an equality sign (=, ≤, or ≥)
  • Represents the status quo or no effect/difference
    • Example: $H_0: \mu = 100$
• The choice of the appropriate type of hypothesis depends on the research question and prior knowledge about the population parameter

One-Sample Tests Explained

• Used when comparing a single sample to a known population parameter
• Aim to determine if the sample mean significantly differs from the hypothesized population mean
• Require assumptions about the population distribution and sample size
  • For normally distributed populations or large sample sizes (n ≥ 30), use the z-test
  • For non-normally distributed populations or small sample sizes (n < 30), use the t-test
• Involve calculating the test statistic using the sample mean, hypothesized population mean, and standard error
  • z-test statistic: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$
  • t-test statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
• The test statistic is then compared to the critical value or used to calculate the p-value
• The decision to reject or fail to reject the null hypothesis is based on the comparison of the p-value to the significance level or the test statistic to the critical value

Common One-Sample Tests

• One-sample z-test:
  • Used when the population standard deviation is known and the sample size is large (n ≥ 30) or the population is normally distributed
  • Example: Testing if the average height of students in a school differs from the national average, given the population standard deviation
• One-sample t-test:
  • Used when the population standard deviation is unknown and the sample size is small (n < 30) or the population is not normally distributed
  • Example: Testing if the average weight of a new product differs from the target weight, using a small sample of products
• One-sample proportion test:
  • Used when comparing a sample proportion to a hypothesized population proportion
  • Example: Testing if the proportion of defective items in a production line differs from the acceptable level
• Chi-square goodness-of-fit test:
  • Used when comparing observed frequencies of categorical data to expected frequencies based on a hypothesized distribution
  • Example: Testing if the observed frequencies of dice rolls match the expected frequencies assuming a fair die

Interpreting Test Results

• If the p-value is less than the significance level or the test statistic falls in the rejection region:
  • Reject the null hypothesis
  • Conclude that there is sufficient evidence to support the alternative hypothesis
  • Example: If p < 0.05, conclude that the sample mean significantly differs from the hypothesized population mean
• If the p-value is greater than or equal to the significance level or the test statistic falls in the non-rejection region:
  • Fail to reject the null hypothesis
  • Conclude that there is not enough evidence to support the alternative hypothesis
  • Example: If p ≥ 0.05, conclude that the sample mean does not significantly differ from the hypothesized population mean
• Interpret the results in the context of the research question and consider the practical significance of the findings
• Be cautious of Type I and Type II errors and consider the power of the test
• Report the test statistic, p-value, and confidence interval (if applicable) along with the decision and interpretation

Real-World Applications

• Quality control:
  • Testing if the mean weight of a product meets the specified requirements
  • Ensuring that the proportion of defective items does not exceed an acceptable level
• Medical research:
  • Comparing the effectiveness of a new treatment to a standard treatment or placebo
  • Determining if a patient's vital signs differ significantly from the normal range
• Psychology:
  • Assessing if a therapy technique leads to a significant improvement in patients' well-being
  • Testing if a personality trait differs between two groups (e.g., introverts and extroverts)
• Business:
  • Evaluating if a marketing campaign has significantly increased sales
  • Comparing the mean customer satisfaction rating to a target value
• Environmental studies:
  • Testing if the average air pollutant levels exceed the legal limits
  • Determining if the proportion of recycled waste meets the city's goals