Justifying a Claim About a Population Mean Based on a Confidence Interval - AP Statistics Study Guide

Quick Summary

This guide will equip you to use a confidence interval as a powerful tool for statistical inference. You will learn how to determine if a claimed value for a population mean is believable by checking if it falls within a calculated interval. Furthermore, you will master the critical relationships between sample size, confidence level, and the width of an interval, enabling you to understand how these factors affect the strength of your conclusions.

Key Concepts

A confidence interval provides a range of plausible values for an unknown population parameter, in this case, the population mean, μ. The core idea of this topic is to use a pre-calculated interval to evaluate a claim.

Using an Interval to Justify a Claim

The logic is simple and direct:

1. Identify the Claim: A claim will propose a specific value for the population mean (e.g., "The average student studies 10 hours per week"). Let's call this claimed value μ₀.

2. Check the Interval: Look at the calculated confidence interval, which has a lower bound and an upper bound.

3. Make a Conclusion:

   • If the claimed value μ₀ is inside the confidence interval, we consider the claim plausible. We do not have convincing statistical evidence to reject the claim.

   • If the claimed value μ₀ is outside the confidence interval, we consider the claim not plausible. We have convincing statistical evidence to reject the claim.

[Image: A number line showing a confidence interval from 8.5 to 11.5. An arrow points to the value 10 inside the interval, labeled "Plausible Claim." Another arrow points to the value 12 outside the interval, labeled "Not a Plausible Claim."]

Example: A 95% confidence interval for the mean study time of all students at a university is (8.5 hours, 11.5 hours).

• Claim 1: The true mean study time is 10 hours.

  • Conclusion: Since 10 is inside the interval (8.5, 11.5), the claim is plausible.

• Claim 2: The true mean study time is 12 hours.

  • Conclusion: Since 12 is outside the interval (8.5, 11.5), the claim is not plausible.

The Anatomy of a Confidence Interval for a Mean

Recall that a confidence interval is constructed as:

Point Estimate ± Margin of Error

For a population mean, this is:

x̄ ± t(sₓ / √n)*

Where:

• x̄ is the sample mean (our point estimate).

• t* is the critical value from the t-distribution with n-1 degrees of freedom.

• sₓ is the sample standard deviation.

• n is the sample size.

• The Margin of Error (ME) is the entire second part: t(sₓ / √n)*. The width of the interval is 2 × ME.

Factors Affecting the Width of a Confidence Interval

Understanding how to make an interval narrower (more precise) or wider (less precise) is crucial.

• Confidence Level:

  • Increasing the confidence level (e.g., from 90% to 99%) makes the interval wider. To be more confident that we have captured the true mean, we need to cast a wider net. This increases the t* critical value, which increases the margin of error.

  • Decreasing the confidence level makes the interval narrower.

• Sample Size (n):

  • Increasing the sample size (n) makes the interval narrower. Larger samples provide more information and reduce variability, leading to a more precise estimate. Mathematically, n is in the denominator of the margin of error formula, so a larger n makes the ME smaller.

  • Decreasing the sample size makes the interval wider.

• Sample Standard Deviation (sₓ):

  • While we cannot control this, it's important to know that a sample with less variability (smaller sₓ) will produce a narrower interval.

Key Vocabulary

• Confidence Interval: An interval of plausible values for an unknown population parameter, calculated from sample data.

• Plausible Value: Any value for a parameter that lies within a calculated confidence interval. It is a value that is not contradicted by the sample data.

• Margin of Error: The value that is added to and subtracted from the point estimate to create the confidence interval. It represents the uncertainty in our estimate.

• Confidence Level: The success rate of the method used to construct the interval. For example, a 95% confidence level means that if we took many samples and built an interval from each, about 95% of those intervals would capture the true parameter.

• Point Estimate: A single value statistic used to estimate a population parameter. For a population mean (μ), the point estimate is the sample mean (x̄).

Calculator Tech (TI-84)

No major new calculator functions are required for this specific topic, which focuses on the interpretation of a given confidence interval.

However, to create the interval that you would then interpret, you would use the $TInterval$ function:

1. Press STAT.

2. Arrow over to the TESTS menu.

3. Select 8:TInterval....

4. Choose your input method: $Data$ (if you have raw data in a list) or $Stats$ (if you are given the sample mean, standard deviation, and sample size).

5. Enter the required values for $\bar{x}$, $Sx$, $n$, and $C-Level$ (confidence level as a decimal).

6. Select $Calculate$. The calculator will output the lower and upper bounds of the confidence interval.

How to Show Work on the FRQ

When an FRQ asks you to use a confidence interval to evaluate a claim, you must write a clear, two-part conclusion. Do not just say "yes" or "no."

FRQ Conclusion Template:

Part 1 (The "Mechanics"): State whether the claimed value is contained within the confidence interval.

"The claimed population mean is μ₀ = [state the claimed value]. This value [is / is not] contained in our calculated [C%] confidence interval of ([lower bound], [upper bound])."

Part 2 (The "Interpretation in Context"): Link the mechanics to your conclusion about the claim, using the word "plausible" or "convincing evidence."

"Because the claimed value [is / is not] in the interval, it [is a plausible value / is not a plausible value] for the true mean [describe the parameter in context]. Therefore, we [do not have / have] convincing statistical evidence to reject the claim that the true mean is [state the claimed value]."

Example using the template:

• Claim: The mean weight of apples is 150 grams.

• Interval: A 95% confidence interval is (152g, 168g).

FRQ Response:

"The claimed population mean is μ₀ = 150 grams. This value is not contained in our calculated 95% confidence interval of (152g, 168g). Because the claimed value is not in the interval, it is not a plausible value for the true mean weight of all apples. Therefore, we have convincing statistical evidence to reject the claim that the true mean weight is 150 grams."

Practice Problems

Problem 1:

A quality control specialist at a bottling plant wants to check if the filling machines are working correctly. The machines are designed to fill bottles with a mean volume of 16.0 ounces. The specialist takes a random sample of 50 bottles and finds the mean volume is 16.08 ounces with a standard deviation of 0.25 ounces. A 95% confidence interval for the true mean fill volume is calculated to be (16.01, 16.15) ounces. Does this interval provide convincing evidence that the machine is not filling bottles to the 16.0 ounce specification? Justify your answer.

Solution:

Using the FRQ template:

The claimed population mean fill volume is μ₀ = 16.0 ounces. This value is not contained in the calculated 95% confidence interval of (16.01, 16.15) ounces.

Because the claimed value of 16.0 ounces is not in the interval, it is not a plausible value for the true mean fill volume of all bottles. Therefore, we have convincing statistical evidence that the machine's true mean fill volume is not 16.0 ounces. The data suggests the machine is overfilling the bottles.

Problem 2:

A researcher calculates a 90% confidence interval for the mean number of hours college students sleep per night, based on a sample of 100 students. The interval is (6.4 hours, 7.2 hours).

(a) A university health center claims that students get, on average, 7 hours of sleep per night. Is this claim plausible based on the interval?

(b) Describe the effect on the width of the interval if the researcher had used a 99% confidence level instead of a 90% confidence level.

(c) Describe the effect on the width of the original 90% interval if the researcher had surveyed 400 students instead of 100.

Solution:

(a) The claimed population mean is μ₀ = 7.0 hours. This value is contained within the 90% confidence interval of (6.4, 7.2) hours. Because the claimed value is in the interval, it is a plausible value for the true mean number of hours college students sleep per night. Therefore, we do not have convincing statistical evidence to reject the health center's claim.

(b) Increasing the confidence level from 90% to 99% would make the interval wider. To be more confident in capturing the true mean, we need a larger margin of error, which results in a wider interval.

(c) Increasing the sample size from 100 to 400 would make the interval narrower. A larger sample size reduces the standard error of the mean (sₓ / √n), which decreases the margin of error and results in a more precise estimate (a narrower interval).

Common Mistakes to Avoid

• Stating a Conclusion Without Justification: Never just say "The claim is plausible." You MUST justify it by stating that the claimed value is inside (or outside) the specific interval. The justification is the most important part of the answer.

• Using "Probability" or "Chance" to Interpret the Interval: Do not say, "There is a 95% chance the true mean is in the interval (16.01, 16.15)." The true mean is a fixed value; it's either in the interval or it's not. The 95% refers to the confidence in the method used to generate the interval, not the probability of a specific interval being correct.

• Confusing "Plausible" with "Proven": Finding a claimed value within an interval does not prove the claim is true. It simply means we don't have enough evidence to say it's false. Statistics is about evidence, not certainty.

• Mixing up the Effects of n and Confidence Level: Remember the key relationships:

  • Higher Confidence Level -> Wider Interval

  • Larger Sample Size (n) -> Narrower Interval

  A common mistake is to think a larger sample size makes us "more confident" and thus makes the interval wider. This is incorrect; a larger sample makes us more precise, leading to a narrower interval.