Statistics: Hypothesis Testing and Sampling

Hypothesis Testing

• A hypothesis is a statement about a statistic that needs to be proven or disproven.
• A hypothesis test is a method to prove a statement as true or false.
• The statement being tested is called the null hypothesis (H₀).
• H₀ is usually a statement of no effect or no difference.
• The alternative hypothesis (Hₐ) is the statement that is tested against the null hypothesis.
• If the data supports rejecting the null hypothesis, it suggests that the alternative hypothesis is likely correct.

Population Proportion and Sample Proportion

• Population: The entire group of interest. A census is conducted to collect data from the entire population.
• Sample: A subset of the population selected for study. Data is collected from this subset using a sample survey.
• Caution: The sample size must be carefully chosen. The sample should not be too small or too large. The sample should also be representative of the population.

Margin of Error

• Margin of Error (ME): A number that represents the amount of sampling error in survey results.
• At a 95% level of confidence: ME = 1/√n, where n is the sample size.
• Larger sample sizes result in smaller margins of error.

Confidence Intervals

• Confidence interval (CI): A range of values that is likely to contain the true population proportion.
• This is calculated using sample proportion and the margin of error
• The CI is calculated at a specified confidence level (example: 95%).
• 95% confidence level means that out of repeated surveys/samples the true population proportion is predicted 95% of the time to fall within the calculated interval
• The confidence interval equation is: Sample proportion - margin of error ≤ Population Proportion ≤ Sample proportion + margin of error.

Standard Error of the Proportion

• The standard error of the sample proportion (σp) is calculated using an improved formula.
• It is a more accurate measure of the variability associated with sample proportion estimates.
• The updated equation is useful when the confidence interval is constructed using the formula p - 1.96(√p(1-p)/n) ≤ P ≤ p+ 1.96(√p(1-p)/n)

Hypothesis Test

• Steps:
  • State the null (H₀) and alternative (Hₐ) hypotheses.
  • Calculate the test statistic (e.g., Z-score or T-score).
  • Determine the p-value.
  • Compare the p-value to the significance level (α = 0.05).
  • If the p-value is less than or equal to α, reject H₀. Otherwise, fail to reject H₀.
• A p-value represents the probability of obtaining results as extreme as or more extreme than those observed, assuming H₀ is true.
• A p-value is used to determine the significance level and whether to fail to reject or reject the null hypothesis

Confidence Intervals (CI) Use in Hypothesis Testing

• A hypothesis test can be performed using this:
  • If the estimated population parameter falls within the calculated CI for the chosen confidence level, you fail to reject the null hypothesis.
  • If the parameter does not fall within this calculated CI, you reject the null hypothesis.

Distributions of Sample Means and the Central Limit Theorem

• The Central Limit Theorem (CLT): If random samples of size n are drawn from any population with a mean of µ and standard deviation of σ, the distribution of sample means will approach a normal distribution as the sample size (n) increases.
• mean of sample means = µ
• standard deviation of sample means = σ/√n

Standard Error

• Standard error (standard deviation of sample means): Is the standard deviation of the sampling distribution of a statistic.
• The formula for the standard error, in the context of the mean of a population, is σ/√n where σ is the standard deviation of the population and n is the sample size.

Test your knowledge on hypothesis testing and sampling methods in statistics. This quiz covers key concepts such as null and alternative hypotheses, population vs sample proportions, and the importance of margin of error. Perfect for students looking to reinforce their understanding of these fundamental statistical principles.