7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval

A statistical claim for population mean is a given population mean from an article, study or other source. For example, looking at the nutritional information on a bag of chicken nuggets, the amount of chicken nuggets per bag is a statistical claim. The company is claiming that the true mean number of chicken nuggets is the listed amount.

If we want to test a statistical claim about a population mean, we have to create a random, independent sample of at least 30 (Central Limit Theorem) and use the mean and standard deviation of that sample to create your confidence interval.

In the chicken nugget example, we would pull at least 30 bags of chicken nuggets randomly and count the number of chicken nuggets in each. Take the mean and standard deviation of our data set to construct our confidence interval.

Remember, we will use t scores to construct our interval since we are estimating a population mean.

When creating a confidence interval, our sample size plays a huge part in our interval.

Our sample size affects two aspects:

Notice that both of these aspects are in the "margin of error" aspect of the confidence interval.

Our critical value for population mean is t score. As you recall from Unit 7.2, our t scores change based on our degrees of freedom (which is based on sample size). As we increase, our sample size, our degrees of freedom will also increase.

For example, if we have a sample size of 41, our degrees of freedom is 40. Looking at our t-chart table (or using your calculator's technology (InvT), we can find that our critical value is 2.021 for a 95% confidence interval.

If we increase our sample size to 51 (df=50), our critical value becomes 2.009.

Therefore, as sample size increases, critical value decreases.

The other aspect of our margin of error that changes with sample size is the standard error. Our standard error formula is found by taking the standard deviation and dividing by the square root of the sample size.

In our example above, if our standard deviation is 1.2, a sample size of 41 yields a standard error of 0.1874

If we increase our sample size to 51, our standard error changes to 0.168

Therefore, as sample size increases, standard error decreases.

Since both aspects of the margin of error decrease with a larger sample size, we can then conclude that the margin of error decreases as a whole when the sample size increases.

In other words, as our sample size increases, our confidence interval gets thinner, better honing in on the population mean that we are trying to estimate.

In order to test the claim of a company or journal article, we look at the range of our confidence interval. If the population mean that is in the claim is contained in our interval, we cannot reject the author's claim. However, if the claimed population mean is NOT in our interval, we have reason to believe that the population mean being claimed could be incorrect. If anything, we have a reason to conduct further studies to test the claim.

In our example with the chicken nuggets, let's say that we find 30 bags of chicken nuggets that have an average of 41.4 chicken nuggets with a standard deviation of 1.2 nuggies.

Let's construct the standard 95% T confidence interval:

point estimate ± (critical value)(standard error)

41.4 ± (2.05)(1.2/√30)

(40.951,41.849)

Looking at our interval above, we can see that the claim from the nutritional facts (40 chicken nuggets) is in fact NOT in our interval. This can lead us to believe that the nutritional information isn't telling the complete truth, but they are in fact giving us EXTRA chicken nuggets! Which makes us 😁😁😁.

And of course, the company is happy because we can't say that we are being cheated out of chicken nuggets. 🙌🙌

WIN....WIN!!

🎥Watch: AP Stats - Inference: Confidence Intervals for Means