Justifying a Claim About a Population Mean Based on a Confidence Interval Chapter Notes | AP Statistics - Grade 9 PDF Download

A statistical claim for the population mean is a statement about the mean or average value of a particular population. This claim is often based on data collected from a sample of the population, and it is used to make inferences about the population as a whole. The population mean is an important measure of central tendency, and it can be used to help understand the characteristics of a population and make predictions about future observations!
Statistical Claim for the Population Mean
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.

Setting Up an Experiment

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.
Remember that the Central Limit Theorem states that the distribution of sample means will be approximately normal, as long as the sample size is large enough (usually n > 30). This means that you can use techniques from inferential statistics, such as constructing a confidence interval, to make inferences about the population mean based on the sample mean.
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.

Importance of Sample Size

When creating a confidence interval, our sample size plays a huge part in our interval.
Our sample size affects two aspects:

1. Critical Value (t)
2. Standard Error

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

Critical Value

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.

Standard Error

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.

Overall Changes

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.

Testing the Claim

In order to test the claim of a company or journal article, we look at the range of our confidence interval. If the claimed population mean is contained within the confidence interval that you have constructed, then you cannot reject the claim. This means that the data you have collected is consistent with the claim, and it is reasonable to accept it as true. ✔️
On the other hand, if the claimed population mean falls outside of the confidence interval, then you have reason to believe that the claim may be incorrect. In this case, you may want to conduct additional studies or gather more data to further 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)

Making a Conclusion

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! Isn't that amazing?
And of course, the company is happy because we can't say that we are being cheated out of chicken nuggets.
Now that's a win-win situation!
Template
“We are C% confident that the confidence interval for a population mean (in context) captures the population mean of ___ (again, in context).”

Key Terms to Review

• Central Limit Theorem: The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution, given that the samples are independent and identically distributed.
• Confidence Interval: A confidence interval is a range of values derived from sample statistics that is likely to contain the true value of an unknown population parameter, with a specified level of confidence.
• Critical Value: A critical value is a point on the scale of the test statistic that is compared to the test statistic to determine whether to reject the null hypothesis.
• Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation without breaking any constraints.
• Hypothesis Testing: Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data.
• Inferences: Inferences are conclusions drawn about a population based on sample data.
• Inferential Statistics: Inferential statistics refers to the branch of statistics that allows us to make conclusions about a population based on a sample of data.
• Margin of Error: The margin of error is a statistical term that quantifies the uncertainty associated with a sample estimate.
• Normal Distribution: Normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve.
• Population Mean: The population mean is the average value of a set of observations for an entire population.
• Standard Error: Standard Error is a statistic that measures the accuracy with which a sample represents a population.
• T Scores: T scores are standardized scores that indicate how many standard deviations a data point is from the mean.