# Hypothesis testing for a mean

Hypothesis testing for a mean is applied for t**esting a value, typically a new finding, against an assumed mean**. As described in Hypothesis testing and in proportion hypothesis testing the null hypothesis is rejected if the new finding falls in the rejection area and is then classified as *significant*.

Hypothesis testing for a mean relates closely to concepts as p-value, significance level (σ), critical value, z-score and test statistics.

## Hypothesis testing for a mean key points

Different ways of explaining hypothesis testing for a mean:

- Hypothesis testing for a mean test
**new finding against an existing assumption**. - The test statistics leads to either
**rejecting or failing to reject**the null hypothesis (H0). - When the new finding falls into the rejection area, it is classified as
**significant**. When it falls into the non-rejection area as**not significant**. - Hypothesis testing answers to the questions: Are the new findings
**significant**or not? Do we**reject**or fail to reject the H0?

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## How to apply hypothesis testing for a mean?

Is the null hypothesis true or not? That’s what we aim to answer with hypothesis testing. Let’s take the example from the Hypothesis testing page:

Our **parrot data analyst** works on an assumed mean of 420 in some procedure that she runs periodically. She conducts a simple random sample with a sample mean of 412.05

Based on the **new sample**, she wishes to test if the assumed mean is less than the 420. The **question** that she would ask is:

*Is 412.05 “significant”** and thus, can we reject the null hypothesis concluding that there is evidence to support the alternative hypothesis stating that the mean is less than 420? *

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## Steps for mean hypothesis testing

For the running of the mean hypothesis testing, an analysis would typically run the procedure below, if she has a value from sample that she wants to analyze and test:

- State the null hypotheses (H0)
- State the alternative hypothesis (HA)
- State significance level (α)
- Calculate test statistics
- Decide

### The hypotheses

As explained in Hypothesis testing, the null hypothesis is the conservative posture claiming that things are as the have always been. The alternative hypothesis is the opposite claiming that our new findings could indicate that there is a change in the value that we use to estimate a certain population parameter.

Our parrot example leads to a null hypothesis saying that our parrots have the same weight or more now that the alternative hypothesis is saying that it is less. Our parrot data analyst would state the following hypotheses:

### Significance level

In our parrot example we could **set the significance level (α) to 0.05 meaning that there is a 5% risk that our data analyst will reject a true null hypothesis**.

The 412.05 should fall within the 0.05 level in order to reject the alternative hypothesis. **In case it falls into the non-rejection area **the null hypothesis would be rejected stating that there is prove to support the alternative hypothesis which states that the mean is less than 420.

### Calculating t statistics

Continuing our parrot example, let’s add a few assumptions for the example:

- The true population is immeasurable (too large to access) and must be estimated with sample statistics
- Weight is a continuous variable, and data is normally distributed
- Sample size (n) < 30

Say that our parrot analyst has taken a **simple random sample of 15 parrots**. The sample mean is 412.05 with a sample standard deviation (s) of 12.21.

To calculate the test score, the statistician will hold the new findings up against the existing, or “old” assumption. **The 412.05 against the 420**. The difference between these two values is seen in relation to the standard error and calculated via the sample statistics:

** **

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### Decide

The p-value of 0.012 tells us that there is a 1.2% chance of finding that extreme a result as the 412.05 that our parrot analyst has found. She will consequently categorize the new finding as significant whereas it is less than the 5% significance level (α) that she set prior to calculating the test statistics.

So, she **rejects the null hypothesis** as there is evidence to support the alternative hypothesis that claims that **the mean is less than the 420**.

## Hypothesis testing for a mean in MS Excel

The Excel function **=STANDARADIZE** calculates the test statistics. Note, that in the argument ‘standard devation’ to enter the standard error for tests where σ are unknown. The **=T-INV** returns the critical t-value, and the **=T.DIST** returns the p-value:

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## Learning statistics

Some of **my preferred learning materials** on hypothesis testing for a mean:

- org (text): Hypothesis testing
- Northern Arizona University (video 13:43): One-sample hypothesis test for a population mean
- JBstatistics (video 13:45): t-Test for One Mean: Introduction
- Khan Academy (video 5:01): Example calculating t statistic for a test about a mean

#### Carsten Grube

Freelance Data Analyst

##### Normal distribution

##### Confidence intervals

##### Simple linear regression, fundamentals

##### Two-sample inference

##### ANOVA & the F-distribution

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