# 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 (H**often states that μ equals a specified value, say μ_{0})_{0}.- The
**Alternative Hypothesis (H**posits that μ does not equal the specified value, or it’s less than or more than μ_{1})_{0}.

**Steps in Conducting a One-sample t-test**

- Step 1: Define and state your
**Null Hypothesis (H**and_{0})**Alternative Hypothesis (H**._{1}) - 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 H**if_{0}**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.