what is the hypothesis of normal distribution

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Normal Distribution Hypothesis Test

Hypothesis tests for the normal distribution can be conducted in a very similar way to binomial distribution , e xcept this time we switch our test statistic. These tests are useful as again they help us test claims of normally distributed items.

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Applied Mathematics
Decision Maths
Discrete Mathematics
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Bias in Experiments
Binomial Distribution
Binomial Hypothesis Test
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How do we carry out a hypothesis test for normal distribution?

When we hypothesis test for the mean of a normal distribution we think about looking at the mean of a sample from a population.

So for a random sample of size n of a population, taken from the random variable \(X \sim N(\mu, \sigma^2)\) , the sample mean \(\bar{X}\) can be normally distributed by \(\bar{X} \sim N(\mu, \frac{\sigma^2}{n})\) .

Let's look at an example.

The weight of crisps is each packet is normally distributed with a standard deviation of 2.5g.

The crisp company claims that the crisp packets have a mean weight of 28g. There were numerous complaints that each crisp packet weighs less than this. Therefore a trading inspector investigated this and found in a sample of 50 crisp packets, the mean weight was 27.2g.

Using a 5% significance level and stating the hypothesis, clearly test whether or not the evidence upholds the complaints.

This is an example of a one tailed test. Let's look at an example of a two tailed test.

A machine produces circular discs with a radius R, where R is normally distributed with a mean of 2cm and a standard deviation of 0.3cm.

The machine is serviced and after the service, a random sample of 40 discs is taken to see if the mean has changed from 2cm. The radius is still normally distributed with a standard deviation of 0.3 cm.

The mean is found to be 1.9cm.

Has the mean changed? Test this to a 5% significance level.

Step 5 may be confusing – do we carry out the calculation with \(P(\bar{X} \leq \bar{x})\) or \(P(\bar{X} \geq \bar{x})\)? As a general rule of thumb if the value is between 0 and the mean, then we use \(P(\bar{X} \leq \bar{x})\) . If it is greater than the mean then we use \(P(\bar{X} \geq \bar{x})\) .

How about finding critical values and critical regions?

This is the same idea as in binomial distribution . However, in normal distribution, a calculator can make our lives easier.

The distributions menu has an option called inverse normal.

Here, we enter the significance level (Area), the mean (\(\mu\) ) and the standard deviation (\(\sigma\) ).

The calculator will give us an answer. Let's have a look at an example below.

Wheels are made to measure for a bike. The diameter of the wheel is normally distributed with a mean of 40cm and a standard deviation of 5cm. Some people think that their wheels are too small. Find the critical value of this to a 5% significance level.

In our calculator, in the inverse normal function, we need to enter:

If we perform the inverse normal function we get 31.775732 .

So that is our critical value and our critical region is \(X \leq 31.775732\) .

Let's look at an example with two tails.

Normal distribution hypothesis test two tailed test studysmarter

Hypothesis Test for Normal Distribution - Key takeaways

When we hypothesis test for a normal distribution we are trying to see if the mean is different from the mean stated in the null hypothesis.
We use the sample mean which is \(\bar{X} \sim N(\mu, \frac{\sigma^2}{n})\) .
In two-tailed tests we divide the significance level by two and test on both tails.
When finding critical values we use the calculator inverse normal function entering the area as the significance level.
For two-tailed tests we need to find two critical values on either end of the distribution.

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Frequently Asked Questions about Normal Distribution Hypothesis Test

How do you test a hypothesis for a normal distribution?

Is hypothesis testing only for a normal distribution?

No, pretty much any distribution can be used when testing a hypothesis. The two distributions that you learn at A-Level are Normal and Binomial.

What statistical hypothesis can be tested in the means of a normal distribution?

We test whether or not the data can support the value of a mean being too low or too high.

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4.1 The normal distribution

Here, you will look at the concept of normal distribution and the bell-shaped curve. The peak point (the top of the bell) represents the most probable occurrences, while other possible occurrences are distributed symmetrically around the peak point, creating a downward-sloping curve on either side of the peak point.

The cartoon shows a bell-shaped curve. The x-axis is titled ‘How high the hill is’ and the y-axis is titled ‘Number of hills’. The top of the bell-shaped curve is labelled ‘Average hill’, but on the lower right tail of the bell-shaped curve is labelled ‘Big hill’.

In order to test hypotheses, you need to calculate the test statistic and compare it with the value in the bell curve. This will be done by using the concept of ‘normal distribution’.

A normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more likely to occur than data far from it. In graph form, a normal distribution appears as a bell curve. The values in the x-axis of the normal distribution graph represent the z-scores. The test statistic that you wish to use to test the set of hypotheses is the z-score . A z-score is used to measure how far the observation (sample mean) is from the 0 value of the bell curve (population mean). In statistics, this distance is measured by standard deviation. Therefore, when the z-score is equal to 2, the observation is 2 standard deviations away from the value 0 in the normal distribution curve.

A symmetrical graph reminiscent of a bell showing normal distribution.

A symmetrical graph reminiscent of a bell. The top of the bell-shaped curve appears where the x-axis is at 0. This is labelled as Normal distribution.