A t test can be used when our data follows a normal distribution. This test looks at the mean of the distribution. This is more commonly used then the z-test since we don’t need to know the standard deviation of the population, rather we can just use the standard deviation of our sample.

For the t-test we also have to look at the degrees of freedom which is found by subtracting 1 from the sample size.

We can use a 1 sided or 2 sided t test.

We can use this test if we want to determine the mean time for your meal to come out at the restaurant with 5 staff on. Our research question is: Is the mean time for our meal to come out at a restaurant 25 minutes?

Step 1: Hypothesis

- H
_{0}is our null hypothesis i.e. the difference between what we observe and what we expected to observe is null and is a result of normal fluctuations.- The mean time for our meal to come out is 25 minutes

- H
_{1}is our alternative hypothesis and assumes that the differences between what we observed and what we expected is of statistical significance.- The mean time for our meal to come out is
**not**25 minutes (2 sided) - The mean time for our meal to come out is greater then 25 minutes (1 sided)

- The mean time for our meal to come out is

Step 2: Analyze evidence

Assumptions

- All data is independent: i.e. we measure the time for multiple meals to come out at different tables throughout the night – we are not just measuring the time for one table’s meals to come out throughout the night.
- The population follows a normal distribution: i.e. the times of different tables can be plotted and follow a normal distribution
- We have to know the standard deviation of all times throughout the night

Test statistic = the (observed mean – expected mean) / (Standard deviation of population/ square root of sample size)

The largest the test statistic, the larger the difference between the observed and expected value hence more evidence that you should reject H_{0}

- So reject H
_{0 }if our t_{n-1}value is less then the significance level - Fail to reject H
_{0}if our t_{n-1}value is greater then the significance level

Note: t_{n-1} will vary for one sided and two sided t tests.

Step 3: Conclusion

Therefore, you can chose to accept or fail to reject H_{0}.

Note: you can never prove anything you just fail to reject it.