📉Intro to Business Statistics Unit 9 – Hypothesis Testing: Single Sample

What's Hypothesis Testing?

• Statistical method used to make inferences about a population based on a sample of data
• Involves formulating a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$)
• Null hypothesis assumes no significant difference or effect exists
• Alternative hypothesis proposes a significant difference or effect is present
• Hypothesis testing helps determine whether to reject or fail to reject the null hypothesis
• Provides a systematic approach to making data-driven decisions in various fields (business, medicine, social sciences)
• Allows for quantifying the level of certainty or uncertainty in the conclusions drawn from the sample data

Types of Hypotheses

• Null hypothesis ($H_0$): Assumes no significant difference, effect, or relationship exists
  • Often represents the status quo or a commonly accepted belief
  • Example: The mean weight of a product is equal to 100 grams ($H_0: \mu = 100$)
• Alternative hypothesis ($H_a$): Proposes a significant difference, effect, or relationship exists
  • Challenges the null hypothesis and suggests an alternative explanation
  • Can be one-sided (greater than or less than) or two-sided (not equal to)
  • Example: The mean weight of a product is greater than 100 grams ($H_a: \mu > 100$)
• Research hypothesis: A specific, testable prediction about the relationship between variables
• Statistical hypothesis: A statement about the parameters of a population distribution

Steps in Hypothesis Testing

1. State the null and alternative hypotheses
   • Clearly define the parameter of interest and the hypothesized value
2. Choose the appropriate test statistic and distribution
   • Depends on the type of data, sample size, and assumptions about the population
3. Specify the significance level ($\alpha$)
   • The probability of rejecting the null hypothesis when it is actually true (Type I error)
4. Collect sample data and calculate the test statistic
   • Use the appropriate formula based on the chosen test and distribution
5. Determine the p-value or critical value
   • P-value: The probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
   • Critical value: The boundary value that separates the rejection and non-rejection regions
6. Make a decision to reject or fail to reject the null hypothesis
   • Compare the p-value to the significance level or the test statistic to the critical value
7. Interpret the results in the context of the research question
   • Discuss the implications and limitations of the findings

Test Statistics and Distributions

• Test statistic: A value calculated from the sample data used to make a decision about the null hypothesis
  • Compares the observed data to what is expected under the null hypothesis
  • Examples: z-statistic, t-statistic, chi-square statistic, F-statistic
• Sampling distribution: The probability distribution of a test statistic under repeated sampling
  • Describes the variability and shape of the test statistic's distribution
  • Depends on the sample size, population distribution, and the null hypothesis
• Normal distribution: A symmetric, bell-shaped distribution used for large sample sizes or known population variance
  • Z-test: Uses the standard normal distribution (mean = 0, standard deviation = 1)
• Student's t-distribution: Similar to the normal distribution but with heavier tails, used for small sample sizes or unknown population variance
  • T-test: Uses the t-distribution with degrees of freedom based on the sample size
• Other distributions: Chi-square, F-distribution, binomial, Poisson, used for specific types of data and hypotheses

Significance Levels and p-values

• Significance level ($\alpha$): The probability of rejecting the null hypothesis when it is actually true (Type I error)
  • Commonly used levels: 0.01, 0.05, 0.10
  • Represents the maximum acceptable level of risk for making a Type I error
• P-value: The probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
  • Measures the strength of evidence against the null hypothesis
  • Smaller p-values provide stronger evidence against the null hypothesis
• Comparing p-value to significance level:
  • If p-value ≤ $\alpha$, reject the null hypothesis
  • If p-value > $\alpha$, fail to reject the null hypothesis
• Relationship between significance level and confidence level:
  • Confidence level = 1 - $\alpha$
  • Example: A significance level of 0.05 corresponds to a 95% confidence level

One-Tailed vs. Two-Tailed Tests

• One-tailed test (directional):
  • Alternative hypothesis specifies a direction (greater than or less than)
  • Tests for a significant difference in only one direction
  • Example: $H_a: \mu > 100$ (population mean is greater than 100)
  • Rejection region is located entirely in one tail of the distribution
• Two-tailed test (non-directional):
  • Alternative hypothesis does not specify a direction (not equal to)
  • Tests for a significant difference in either direction
  • Example: $H_a: \mu \neq 100$ (population mean is not equal to 100)
  • Rejection region is divided equally between both tails of the distribution
• Choosing between one-tailed and two-tailed tests:
  • Depends on the research question and prior knowledge about the direction of the effect
  • One-tailed tests are more powerful but require a strong justification for the direction
  • Two-tailed tests are more conservative and appropriate when the direction is unknown or not specified

Common Single Sample Tests

• One-sample z-test:
  • Used when the population standard deviation is known and the sample size is large (n ≥ 30)
  • Tests hypotheses about the population mean
  • Example: Testing if the average height of students differs from the national average
• One-sample t-test:
  • Used when the population standard deviation is unknown and the sample size is small (n < 30)
  • Tests hypotheses about the population mean
  • Example: Testing if a new teaching method improves student performance compared to a historical average
• One-sample proportion test:
  • Used when the data is categorical and the sample size is large (np ≥ 10 and n(1-p) ≥ 10)
  • Tests hypotheses about the population proportion
  • Example: Testing if the proportion of defective products exceeds a specified threshold
• Chi-square goodness-of-fit test:
  • Used when the data is categorical and the expected frequencies are known
  • Tests if the observed frequencies differ significantly from the expected frequencies
  • Example: Testing if the distribution of colors in a bag of M&Ms matches the company's claims

Interpreting Results and Making Decisions

• Statistical significance:
  • Rejecting the null hypothesis indicates a statistically significant result
  • Suggests that the observed difference or effect is unlikely to have occurred by chance alone
  • Does not necessarily imply practical or clinical significance
• Practical significance:
  • Considers the magnitude and relevance of the effect in the context of the research question
  • Assesses whether the difference or effect is large enough to be meaningful in real-world applications
• Type I error (false positive):
  • Rejecting the null hypothesis when it is actually true
  • Controlled by the significance level ($\alpha$)
• Type II error (false negative):
  • Failing to reject the null hypothesis when it is actually false
  • Related to the power of the test (1 - $\beta$)
• Confidence intervals:
  • Provide a range of plausible values for the population parameter
  • Reflect the uncertainty associated with the sample estimate
  • Can be used to assess the precision and significance of the results
• Making decisions based on hypothesis testing:
  • Consider both statistical and practical significance
  • Interpret the results in the context of the research question and prior knowledge
  • Acknowledge the limitations and potential sources of bias in the study design and data collection
  • Use the findings to inform future research and decision-making processes