Understanding Hypothesis Testing for Normal Distribution

Hypothesis testing is a useful tool for validating claims about normally distributed items, similar to its application in binomial distribution. However, there is a slight variation in the test statistic. But how exactly do we conduct a hypothesis test for normal distribution?

When testing the mean of a normal distribution, we focus on the mean of a sample from a population. In simpler terms, for a random sample of size n from a population with a random variable X, the sample mean is represented by X̄ ~ N(µ, σ²/n). To better grasp this concept, let’s look at an example.

Let’s say a crisp company claims that each packet of crisps has a mean weight of 28g, with a standard deviation of 2.5g. However, after receiving multiple complaints regarding the weight of the crisps being less than 28g, a trading inspector decides to investigate. They take a sample of 50 crisp packets and find that the mean weight is 27.2g. Using a 5% significance level, let's test whether or not the evidence supports the complaints.

This is considered a one-tailed test, where the alternative hypothesis is that the mean weight is less than 28g. On the other hand, a two-tailed test may also be conducted. For example, a machine is serviced, and a random sample of 40 circular discs is taken to see if the mean radius has changed from 2cm. The radius is normally distributed with a mean of 2cm and a standard deviation of 0.3cm. The sample mean is found to be 1.9cm. Is there enough evidence to suggest a change in the mean? Let’s test this with a 5% significance level.

The fifth step in conducting a hypothesis test can be a little confusing – do we use σ²/n or just σ²? A simple rule of thumb is to use σ²/n if the calculated test statistic falls between 0 and the mean, and to use σ² if it is greater than the mean.

Now, let’s talk about finding critical values and critical regions. This follows the same concept as in binomial distribution, but with normal distribution, we can use calculators to simplify the process. Most calculator menus have an option called "inverse normal," where we can input the significance level (Area), mean (µ), and standard deviation (σ) to obtain the critical value. For instance, if we want to find the critical value for a significance level of 5%, we enter 0.05 as the Area.

Let’s see this in action with an example. The diameter of wheels for a bike is normally distributed with a mean of 40cm and a standard deviation of 5cm. Some people think that their wheels are too small. What is the critical value for a 5% significance level? Using the inverse normal function on our calculator with the given values, we get a critical value of 39.289. This suggests that any sample mean below 39.289 would fall in the critical region, providing evidence for the alternative hypothesis.

In conclusion, when conducting a hypothesis test for normal distribution, we are essentially trying to determine if the mean is different from the mean stated in the null hypothesis. We use the sample mean as our test statistic and divide the significance level by two for two-tailed tests. With the help of calculators, finding critical values and regions can be done more conveniently. Therefore, whether it is a normal or binomial distribution, hypothesis testing can be applied to any distribution to either support or reject a statistical hypothesis about the mean.