What is the process for conducting a left-tailed hypothesis test on a population mean in statistics?

Statistics – Hypothesis Testing a Mean (Left Tailed)

A population mean is an average of value a population.

Hypothesis tests are used to check a claim about the size of that population mean.

Hypothesis Testing a Mean

The following steps are used for a hypothesis test:

1. Check the conditions
2. Define the claims
3. Decide the significance level
4. Calculate the test statistic
5. Conclusion

For example:

• Population: Nobel Prize winners
• Category: Age when they received the prize.

And we want to check the claim:

By taking a sample of 30 randomly selected Nobel Prize winners we could find that:

From this sample data we check the claim with the steps below.

1. Checking the Conditions

The conditions for calculating a confidence interval for a proportion are:

• The sample is randomly selected
• And either:
  • The population data is normally distributed
  • Sample size is large enough

A moderately large sample size, like 30, is typically large enough.

In the example, the sample size was 30 and it was randomly selected, so the conditions are fulfilled.

2. Defining the Claims

We need to define a null hypothesis ((H_{0})) and an alternative hypothesis ((H_{1})) based on the claim we are checking.

The claim was:

In this case, the parameter is the mean age of Nobel Prize winners when they received the prize ((mu)).

The null and alternative hypothesis are then:

Which can be expressed with symbols as:

This is a ‘left tailed’ test, because the alternative hypothesis claims that the proportion is less than in the null hypothesis.

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

3. Deciding the Significance Level

The significance level ((alpha)) is the uncertainty we accept when rejecting the null hypothesis in a hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

• (alpha = 0.1) (10%)
• (alpha = 0.05) (5%)
• (alpha = 0.01) (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no “correct” significance level – it only states the uncertainty of the conclusion.

4. Calculating the Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

The formula for the test statistic (TS) of a population mean is:

(displaystyle frac{bar{x} – mu}{s} cdot sqrt{n} )

(bar{x}-mu) is the difference between the sample mean ((bar{x})) and the claimed population mean ((mu)).

(s) is the sample standard deviation.

(n) is the sample size.

In our example:

So the test statistic (TS) is then:

(displaystyle frac{62.1-60}{13.46} cdot sqrt{30} = frac{2.1}{13.46} cdot sqrt{30} approx 0.156 cdot 5.477 = underline{0.855})

You can also calculate the test statistic using programming language functions:

5. Concluding

There are two main approaches for making the conclusion of a hypothesis test:

• The critical value approach compares the test statistic with the critical value of the significance level.
• The P-value approach compares the P-value of the test statistic and with the significance level.

The Critical Value Approach

For the critical value approach we need to find the critical value (CV) of the significance level ((alpha)).

For a population mean test, the critical value (CV) is a T-value from a student’s t-distribution.

This critical T-value (CV) defines the rejection region for the test.

The rejection region is an area of probability in the tails of the standard normal distribution.

Because the claim is that the population mean is less than 60, the rejection region is in the left tail:

The size of the rejection region is decided by the significance level ((alpha)).

The student’s t-distribution is adjusted for the uncertainty from smaller samples.

This adjustment is called degrees of freedom (df), which is the sample size ((n) – 1)

In this case the degrees of freedom (df) is: (30 – 1 = underline{29} )

Choosing a significance level ((alpha)) of 0.05, or 5%, we can find the critical T-value from a T-table, or with a programming language function:

Using either method we can find that the critical T-Value is (approx underline{-1.699})

For a left tailed test we need to check if the test statistic (TS) is smaller than the critical value (CV).

If the test statistic is smaller the critical value, the test statistic is in the rejection region.

When the test statistic is in the rejection region, we reject the null hypothesis ((H_{0})).

Here, the test statistic (TS) was (approx underline{0.855}) and the critical value was (approx underline{-1.699})

Here is an illustration of this test in a graph:

Since the test statistic was bigger than the critical value we keep the null hypothesis.

This means that the sample data does not support the alternative hypothesis.

And we can summarize the conclusion stating:

The P-Value Approach

For the P-value approach we need to find the P-value of the test statistic (TS).

If the P-value is smaller than the significance level ((alpha)), we reject the null hypothesis ((H_{0})).

The test statistic was found to be ( approx underline{0.855} )

For a population proportion test, the test statistic is a T-Value from a student’s t-distribution.

Because this is a left tailed test, we need to find the P-value of a t-value smaller than 0.855.

The student’s t-distribution is adjusted according to degrees of freedom (df), which is the sample size ((30) – 1 = underline{29})

We can find the P-value using a T-table, or with a programming language function:

Using either method we can find that the P-value is (approx underline{0.800})

This tells us that the significance level ((alpha)) would need to be smaller 0.80, or 80%, to reject the null hypothesis.

Here is an illustration of this test in a graph:

This P-value is far bigger than any of the common significance levels (10%, 5%, 1%).

So the null hypothesis is kept at all of these significance levels.

And we can summarize the conclusion stating:

Calculating a P-Value for a Hypothesis Test with Programming

Many programming languages can calculate the P-value to decide outcome of a hypothesis test.

Using software and programming to calculate statistics is more common for bigger sets of data, as calculating manually becomes difficult.

The P-value calculated here will tell us the lowest possible significance level where the null-hypothesis can be rejected.

Left-Tailed and Two-Tailed Tests

This was an example of a left tailed test, where the alternative hypothesis claimed that parameter is smaller than the null hypothesis claim.

You can check out an equivalent step-by-step guide for other types here:

• Right-Tailed Test
• Two-Tailed Test