Statistics — Hypothesis Testing

“A thesis is something that has been proven to be true. A hypothesis is something that has not yet been proven to be true.”

  1. The null hypothesis

The procedure of hypothesis testing aims to determine whether the given hypothesis is true or not through statistical methods. The first step of hypothesis test is to set up the null hypothesis. A null hypothesis is an assertion about the value of a population parameter. It is considered to be true until we have sufficient statistical evidence to reject. For example, a delivery vendor claims that his company could deliver parcels, on the average, in at most 3 working days. You suspect that the average is greater than 3 working days and would like to test it. The null hypothesis would be set up as following:

We put what we would like to reject in the null hypothesis (we usually put equal sign in the null hypothesis), we use data and statistical procedures to reject or fail to reject the null hypothesis. The alternative hypothesis is the negation of the null hypothesis, thus, in this case, the alternative hypothesis would be set up as following:

The null hypothesis and the alternative hypothesis are exactly opposite statements, thus, only one of them could be true. However, we could never be one hundred percentage certain that whether we are right in rejecting or failing to reject the null hypothesis.

2. Type I and Type II error

As shown in the table, there are total two kinds of error situations could occur while testing. A Type I error means rejecting a null hypothesis when it is true, a Type II error means fail to reject a null hypothesis when it is false. A significance level is always attached while doing the test, a significance level is the probability of committing Type I error. A 5% significance level corresponds to 95% confidence interval. We could reduce the probability of committing Type I error by lowering the significance level when testing, however, meanwhile the probability of committing Type II error will be increasing.

3. Hypothesis testing about the mean

Before we doing the test we should distinguish between large samples and small samples, i.e. using the values of a z table or t-distribution.

Large samples: Testing about the mean (known the population standard deviation)

Example: A chocolate manufacturer sold in 200g packs. We want to put a claim that the weights of the packs actually distributed over 200g, from previous data we know that the standard deviation of the weights of the packs is 18g. We take a random sample of 36 packs and find the sample mean 203g.

If our test statistic is below the critical value (for lower-tail tests) or above the critical value (for upper-tail tests), then we reject the null hypothesis.

Small samples: Testing about the mean (unknown the population standard deviation)

Because the population standard deviation is unknown (most cases in real-world, the population standard deviation is unknown), thus, we could use an estimation, the sample standard deviation instead of population standard deviation in the formula of z statistic.

The statistic no longer has a standard normal distribution, the statistic follows a t distribution:

4. Hypothesis testing about the variance

Example: The chocolate manufacturer want to test (reject) the population variance of the weights of the packs is greater than 100. We take a random sample of 36 packs and find the sample variance is 81.

5. Conclusion

The steps of hypothesis testing:

1- Formulate the null hypothesis and the alternative hypothesis, we put the claim we would like to reject in the null hypothesis.

2- Calculate the test statistics.

3- Check the table of the values for the critical value and decide whether to reject or fail to reject the null hypothesis.

Reference

Aczel, A. D. and Sounderpandian, J. (2009) Complete Business Statistics. New York: McGraw-Hill/Irwin.

Emvalomatis, G. (2020) ‘HypothesisTesting’ [PowerPoint presentation]. BU52018: Business Analytics.

Ozdemir, D. (2016) Applied Statistics for Economics and Business. Switzerland: Springer.

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