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

4 min readjanuary 4, 2023

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

A statistical claim for the 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 about the population as a whole. The 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!

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. 🐔

https://i5.walmartimages.com/asr/b65efd59-ee8c-4d40-8e79-c9213f89d25c_1.607981dffc2aebdc4d04e4257d6c475d.png?odnWidth=undefined&odnHeight=undefined&odnBg=ffffff

image courtesy of: walmart.com

Setting Up an Experiment

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

Remember that the 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 , such as constructing a , to make about the 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 .

Remember, we will use to construct our interval since we are estimating a .

Importance of Sample Size

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

Our sample size affects two aspects:

  1. (t*)

Notice that both of these aspects are in the "" aspect of the .

Critical Value

Our for is t score. As you recall from Unit 7.2, our change based on our (which is based on sample size). As we increase, our sample size, our will also increase.

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

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

Therefore, as sample size increases, decreases.

Standard Error

The other aspect of our that changes with sample size is the . Our 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 of 0.1874

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

Therefore, as sample size increases, decreases.

Overall Changes

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

In other words, as our sample size increases, our gets thinner, better honing in on the 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 . If the claimed is contained within the 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 falls outside of the , 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 :

point estimate ± ()()

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 for a (in context) captures the of ___ (again, in context)."

https://i.kym-cdn.com/photos/images/original/001/652/891/0fe.jpg

image courtesy of: knowyourmeme.com

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

Key Terms to Review (12)

Central Limit Theorem

: The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution.

Confidence Interval

: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.

Critical Value

: A critical value is a specific value that separates the rejection region from the non-rejection region in hypothesis testing. It is compared to the test statistic to determine whether to reject or fail to reject the null hypothesis.

Degrees of Freedom

: Degrees of freedom refers to the number of values in a calculation that are free to vary. In statistics, it represents the number of independent pieces of information available for estimating a parameter.

Hypothesis Testing

: Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, collecting data, calculating test statistics, and making decisions about rejecting or failing to reject the null hypothesis.

Inferences

: Inferences in statistics refer to drawing conclusions or making predictions about a population based on sample data.

Inferential Statistics

: Inferential statistics involves using sample data to make inferences or draw conclusions about a population.

Margin of Error

: The margin of error is a measure of the uncertainty or variability in survey results. It represents the range within which the true population parameter is likely to fall.

Normal Distribution

: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.

Population Mean

: The population mean is the average value of a variable for an entire population. It represents a summary measure for all individuals or units within that population.

Standard Error

: The standard error is a measure of the variability or spread of sample means around the population mean. It tells us how much we can expect sample means to differ from the true population mean.

T Scores

: T scores are standardized scores that represent how far an individual's score deviates from the mean in units of standard deviation when working with small samples.

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

4 min readjanuary 4, 2023

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

A statistical claim for the 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 about the population as a whole. The 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!

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. 🐔

https://i5.walmartimages.com/asr/b65efd59-ee8c-4d40-8e79-c9213f89d25c_1.607981dffc2aebdc4d04e4257d6c475d.png?odnWidth=undefined&odnHeight=undefined&odnBg=ffffff

image courtesy of: walmart.com

Setting Up an Experiment

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

Remember that the 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 , such as constructing a , to make about the 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 .

Remember, we will use to construct our interval since we are estimating a .

Importance of Sample Size

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

Our sample size affects two aspects:

  1. (t*)

Notice that both of these aspects are in the "" aspect of the .

Critical Value

Our for is t score. As you recall from Unit 7.2, our change based on our (which is based on sample size). As we increase, our sample size, our will also increase.

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

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

Therefore, as sample size increases, decreases.

Standard Error

The other aspect of our that changes with sample size is the . Our 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 of 0.1874

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

Therefore, as sample size increases, decreases.

Overall Changes

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

In other words, as our sample size increases, our gets thinner, better honing in on the 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 . If the claimed is contained within the 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 falls outside of the , 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 :

point estimate ± ()()

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 for a (in context) captures the of ___ (again, in context)."

https://i.kym-cdn.com/photos/images/original/001/652/891/0fe.jpg

image courtesy of: knowyourmeme.com

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

Key Terms to Review (12)

Central Limit Theorem

: The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution.

Confidence Interval

: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.

Critical Value

: A critical value is a specific value that separates the rejection region from the non-rejection region in hypothesis testing. It is compared to the test statistic to determine whether to reject or fail to reject the null hypothesis.

Degrees of Freedom

: Degrees of freedom refers to the number of values in a calculation that are free to vary. In statistics, it represents the number of independent pieces of information available for estimating a parameter.

Hypothesis Testing

: Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, collecting data, calculating test statistics, and making decisions about rejecting or failing to reject the null hypothesis.

Inferences

: Inferences in statistics refer to drawing conclusions or making predictions about a population based on sample data.

Inferential Statistics

: Inferential statistics involves using sample data to make inferences or draw conclusions about a population.

Margin of Error

: The margin of error is a measure of the uncertainty or variability in survey results. It represents the range within which the true population parameter is likely to fall.

Normal Distribution

: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.

Population Mean

: The population mean is the average value of a variable for an entire population. It represents a summary measure for all individuals or units within that population.

Standard Error

: The standard error is a measure of the variability or spread of sample means around the population mean. It tells us how much we can expect sample means to differ from the true population mean.

T Scores

: T scores are standardized scores that represent how far an individual's score deviates from the mean in units of standard deviation when working with small samples.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.