#AP Statistics: Confidence Intervals for Population Means 🚀

Hey there, future AP Stats superstar! Let's get you prepped and confident for the exam. We're diving into confidence intervals for population means, a topic that's super important and often shows up in different forms on the test. This guide will be your best friend tonight, so let's make every minute count!

#Understanding Statistical Claims About Population Means

A statistical claim for the population mean is a statement about the average value of a particular population. Think of it like a company's claim about the average number of chicken nuggets in a bag. We use sample data to make inferences about the entire population. It's all about using what we know from a small group to say something about a much larger group. 🐔

Image courtesy of: walmart.com

#Setting Up Your Experiment 🧪

To test a claim about a population mean, we need a random, independent sample of at least 30. This is where the Central Limit Theorem (CLT) comes in! It tells us that the distribution of sample means will be approximately normal if our sample size is large enough (n ≥ 30). This allows us to use t-distributions to construct our confidence interval.

In the chicken nugget example, we'd randomly grab at least 30 bags, count the nuggets in each, and then calculate the mean and standard deviation of our sample. We'll use these to build our confidence interval using t-scores since we are estimating a population mean.

#The Impact of Sample Size 🧠

Sample size is a big deal! It affects both the critical value (t)* and the standard error, which are both part of the margin of error. Let's see how:

#Critical Value (t*)

The critical value (t*) depends on the degrees of freedom (df), which is based on our sample size (n-1). As sample size increases, degrees of freedom increase, and the t* value decreases.

For example:

• If n = 41, df = 40, and t* ≈ 2.021 (for 95% confidence)

• If n = 51, df = 50, and t* ≈ 2.009 (for 95% confidence)

So, as sample size increases, the critical value decreases.

#Standard Error

The standard error is calculated by dividing the sample standard deviation by the square root of the sample size: $SE = \frac{s}{\sqrt{n}}$

Using our example with a standard deviation of 1.2:

• If n = 41, SE ≈ 0.1874

• If n = 51, SE ≈ 0.168

So, as sample size increases, standard error decreases.

#Overall Changes

Since both t* and SE decrease with larger sample sizes, the overall margin of error decreases. This means that as our sample size increases, our confidence interval becomes narrower, giving us a more precise estimate of the population mean.

#Testing the Claim 🎯

To test a claim, check if the claimed population mean falls within your confidence interval.

• If the claimed mean is inside the interval: You can't reject the claim. Your data is consistent with the claim. ✔️

• If the claimed mean is outside the interval: You have reason to doubt the claim. Further investigation may be needed.

Let's go back to our chicken nugget example. Suppose we found that 30 bags have a sample mean of 41.4 nuggets with a standard deviation of 1.2. Let's construct a 95% confidence interval:

point estimate ± (critical value)(standard error)

$41.4 ± (2.042)(\frac{1.2}{\sqrt{30}}) = (40.951, 41.849)$

#Making a Conclusion 🎉

In our example, the company's claim of 40 nuggets is not within our interval (40.951, 41.849). This suggests that the bags might contain more nuggets than advertised!

#Template

"We are C% confident that the confidence interval for a population mean (in context) captures the population mean of ___ (again, in context)."

For our example:

"We are 95% confident that the confidence interval (40.951, 41.849) captures the true population mean number of chicken nuggets per bag."

Image courtesy of: knowyourmeme.com

#Final Exam Focus 🎯

• Key Topics: Confidence intervals for population means, Central Limit Theorem, t-distributions, impact of sample size.

• Common Question Types: Constructing and interpreting confidence intervals, determining if a claim is supported by data, understanding the relationship between sample size and interval width.

• Time Management: Practice constructing intervals quickly and accurately. Focus on understanding the concepts rather than just memorizing formulas.

• Common Pitfalls: Using z-scores instead of t-scores, misinterpreting the meaning of a confidence interval, not understanding the impact of sample size.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A researcher wants to estimate the average height of adult women in a city. They take a random sample of 100 women and calculate a 95% confidence interval for the mean height. If they wanted to reduce the width of the confidence interval, which of the following would be most effective? (A) Increase the sample size to 400 (B) Decrease the sample size to 25 (C) Increase the confidence level to 99% (D) Use a t-distribution instead of a z-distribution

2. A company claims that the average battery life of their smartphones is 12 hours. A consumer group takes a random sample of 64 smartphones and finds a sample mean battery life of 11.5 hours with a standard deviation of 2 hours. They construct a 95% confidence interval for the population mean. Which of the following is a correct interpretation of the interval? (A) There is a 95% chance that the true population mean is within the calculated interval. (B) We are 95% confident that the true population mean is within the calculated interval. (C) We are 95% confident that the sample mean is within the calculated interval. (D) There is a 95% chance that the sample mean is within the calculated interval.

#Free Response Question

A local bakery claims that the average weight of their chocolate chip cookies is 2.5 ounces. A customer suspects that the cookies are actually lighter than claimed. They randomly select 40 cookies and weigh each one, finding a sample mean weight of 2.35 ounces with a sample standard deviation of 0.4 ounces.

(a) Construct a 95% confidence interval for the true mean weight of the bakery's chocolate chip cookies. (b) Based on your confidence interval, is there sufficient evidence to suggest that the bakery's claim is incorrect? Explain your reasoning. (c) Suppose the customer had taken a sample of 100 cookies instead of 40. How would this change the width of the confidence interval? Explain.

#FRQ Scoring Guidelines

(a) Construct a 95% confidence interval for the true mean weight of the bakery's chocolate chip cookies.

• Step 1: Identify the correct formula and conditions
  • Formula: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$
  • Conditions: Random sample, n ≥ 30 (or population is approximately normal)
• Step 2: Calculate the critical value
  • Degrees of freedom: df = 40 - 1 = 39
  • t* ≈ 2.023 (using t-table or calculator)
• Step 3: Calculate the standard error
  • $SE = \frac{0.4}{\sqrt{40}} ≈ 0.063$
• Step 4: Calculate the interval
  • 2.35 ± (2.023)(0.063) = (2.222, 2.478)
• Step 5: Correct interpretation
  • We are 95% confident that the true mean weight of the chocolate chip cookies is between 2.222 and 2.478 ounces.

(b) Based on your confidence interval, is there sufficient evidence to suggest that the bakery's claim is incorrect? Explain your reasoning.

• Since the claimed mean of 2.5 ounces is not within the 95% confidence interval (2.222, 2.478), there is sufficient evidence to suggest that the bakery's claim is incorrect. The data suggests that the true mean weight of the cookies is likely less than 2.5 ounces.

(c) Suppose the customer had taken a sample of 100 cookies instead of 40. How would this change the width of the confidence interval? Explain.

• Increasing the sample size to 100 would decrease the standard error, and therefore the width of the confidence interval. A larger sample size provides more information about the population, leading to a more precise estimate of the population mean. The critical value (t*) would also decrease slightly, further reducing the width of the interval.

You've got this! Remember to stay calm, trust your preparation, and tackle each question step-by-step. You're ready to ace this exam! 💪