Hypothesis Testing for a Mean Using Normal Distribution

Hypothesis Testing for a Mean Using Normal Distribution

Normal Distribution Basics

  • A normal distribution, often referred to as a bell curve, models continuous data where the mean and median are equal and located at the centre of the distribution.
  • The standard normal distribution has a mean of 0 and standard deviation of 1.
  • To conduct hypothesis tests related to the mean, the distribution of the population must be approximately normal.

Hypothesis Testing using the Normal Distribution

  • The main aim of conducting a one-sample t-test is to test whether the population mean, μ, is equal to a specified value.
  • Null Hypothesis (H0) often states that μ equals a specified value, say μ0.
  • The Alternative Hypothesis (H1) posits that μ does not equal the specified value, or it’s less than or more than μ0.

Steps in Conducting a One-sample t-test

  • Step 1: Define and state your Null Hypothesis (H0) and Alternative Hypothesis (H1).
  • Step 2: Choose your significance level (α). It is often set at 0.05.
  • Step 3: Calculate the test statistic (t). This is computed by (sample mean - hypothesized mean) / (sample standard deviation / square root of sample size).
  • Step 4: Find the p-value associated with the observed value of the test statistic.
  • Step 5: Compare the p-value with the significance level α. A p-value less than α gives strong evidence against the null hypothesis and thus it can be rejected.

Interpreting the Result of the One-sample t-test

  • Reject H0 if p-value is less than the significance level. This shows significant evidence against the null hypothesis.
  • If p-value is more than the significance level, then the null hypothesis cannot be rejected. This indicates that there isn’t sufficient evidence against it.

Assumptions for one-sample t-tests

  • The one-sample t-test assumes that the population from which the sample is drawn is normally distributed.
  • All items in the sample are independently drawn from the population.
  • The variance of the population is unknown.

Error Types in one-sample t-tests

  • Type I error is rejecting the null hypothesis when it is true, resulting in a false positive.
  • Type II error is failing to reject the null hypothesis when it is false, resulting in a false negative.