Below, we have imported the Python libraries needed for this module. Run the code in this cell before running any other code cells, and be careful not to change any of the code. You can run the cell in any of these ways:

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Estimated Time: ~45 minutes
Professor: Alice Martinez
Developer: Rinrada Maneenop

Content Overview¶

• Introduction: What is a Confidence Interval?

• Confidence Intervals for a Population Mean

• Bootstrap Confidence Intervals

• Using Confidence Intervals to Test Hypotheses

• Interactive: How Sample Size and Confidence Level Affect Width

• Discussion and Reflection

• Conclusion

Introduction: What is a Confidence Interval?¶

A confidence interval is a way to use data from a random sample to estimate an unknown population parameter, such as a population mean or proportion.

Instead of giving a single guess, we report a range of plausible values based on our sample. For example, we might say:

“We are 95% confident that the average number of hours community college students study per week is between 9.2 and 11.8 hours.”

In this statement:

• The parameter we care about is the true average study time for all students in the population.

• The interval (9.2, 11.8) is built using sample data and probability rules from statistics.

It is important to remember:

• The interval estimates the parameter, not individual data values.

• The confidence level (like 95%) describes the method, not a single interval. If we repeated the whole study many times, about 95% of the resulting intervals would contain the true parameter.

A Common Misunderstanding¶

Suppose we build a 95% confidence interval for the average age of patients at a clinic and get something like (27.1, 29.4) years.

It is not correct to say:

“95% of individual patients are between 27.1 and 29.4 years old.”

That would confuse the population distribution of ages (which might be very spread out) with the uncertainty in our estimate of the mean.

What the interval does tell us is:

• We used a method that, in the long run, captures the true mean age about 95% of the time.

• For this particular sample, our best guess is that the true mean age is somewhere between 27.1 and 29.4 years.

So the correct way to say this is:

We are 95% confident that the true mean (average) age of patients at the clinic is between 27.1 and 29.4 years old

Confidence Intervals for a Population Mean¶

In this section we will:

• Work with a sample of numerical data.

• Build a confidence interval for the population mean using the standard normal distribution from scipy.stats.

• Interpret the interval in plain language.

Formula for a Confidence Interval for the Mean¶

For a sample of size $n$ with sample mean $\bar{x}$ and sample standard deviation $s$, an approximate $100(1-\alpha)\%$ confidence interval for the population mean $\mu$ is:
NOTE: $\alpha$ is the significance level, where the interval does not contain the true mean

where:

• $\bar{x}$ - sample mean

• $n$ - sample size

• $s$ - sample standard deviation

• $z$ - The critical value from the standard normal distribution. This value determines how far from the mean you need to go to capture the desired confidence level.

• Standard Error ($\frac{s}{\sqrt{n}}$): Estimates how much the sample mean $\bar{x}$ would vary across repeated samples of size $n$.

We will use scipy.stats.norm to get (z) and then compute the interval in code.

Two-tailed vs one-tailed¶

The formula above gives a two-tailed (two-sided) interval: we allow the true mean to be either above or below our sample mean, so we put half of α in each tail of the distribution. That’s why the code uses 1 - alpha/2 to get (z): we want 2.5% in the left tail and 2.5% in the right tail for a 95% CI. The result is an interval with both a lower and an upper bound: “the mean is between this and this.”

• One-tailed would mean we care about only one direction (e.g. “is the mean greater than some value?”). We’d put all of α in a single tail and get only one bound (either a lower bound or an upper bound), not a full interval.

In this notebook we use two-tailed intervals so we can say “the mean is between [lower] and [upper].”

Exercise 1: Fill in the ... in the next cell to compute the 95% confidence interval using the formula $\bar{x} \pm z \cdot \frac{s}{\sqrt{n}}$.

Interpreting the Interval¶

If this sample came from all community college students at your school, we might say:

“Based on our sample, we are 95% confident that the average number of hours students study per week is between the lower and upper endpoints we just computed.”

This does not mean that there is a 95% chance that the true mean is in this exact interval. Instead, it means that we are using a method that, in the long run, captures the true mean about 95% of the time.

Bootstrap Confidence Intervals¶

The formula above relies on theoretical assumptions (for example, that the sampling distribution of the mean is roughly bell-shaped). When the sample size is small or the data are clearly skewed, a bootstrap confidence interval can be a helpful alternative.

The idea of the bootstrap is:

• Treat your original sample as if it were the population.

• Resample with replacement from the sample many times.

• For each resample, compute the statistic of interest (here, the mean).

• Use the distribution of these bootstrap statistics to build a confidence interval.

Exercise 2: Complete the bootstrap_mean function (resample and mean) and the np.percentile call for the 95% bootstrap CI.

The vertical red lines mark the endpoints of a 95% bootstrap confidence interval for the population mean study time.

Notice that this interval is built using the empirical distribution of the bootstrap means, not directly from a theoretical t distribution. Both methods are implemented using the same core tools you have seen elsewhere in this course: numpy for sampling and scipy.stats for probability calculations.

Using Confidence Intervals to Test Hypotheses¶

Confidence intervals are not only useful for estimation; they are also closely connected to hypothesis tests.

Suppose we want to test:

• Null hypothesis (H_0): The average drop in a health score after treatment is (0) (no change on average).

• Alternative hypothesis (H_1): The average drop is not 0.

If a 95% confidence interval for the mean drop does not contain 0, then a two-sided hypothesis test at the 5% significance level would reject (H_0). This is the practical “duality” between confidence intervals and two-sided tests.

Reading the plot: The salmon-colored smooth curve (the KDE, or kernel density estimate) shows the shape of the distribution of drop scores. It smooths the histogram so you can more easily see where values are concentrated and whether the distribution is roughly symmetric or skewed. The black dotted vertical line is the sample mean drop in health score—the average change across all patients in the sample.

Look at the interval endpoints you just computed.

• If 0 is inside the 95% bootstrap confidence interval for the mean drop, then a two-sided test at the 5% level would not find strong evidence that the average drop is different from 0.

• If 0 is outside the interval, then a two-sided test at the 5% level would reject the null hypothesis that the average drop is 0.

In practice, checking whether the null value lies in a 95% confidence interval is often easier and more informative than computing a separate test statistic and p-value.

Interactive: Sample Size, Confidence Level, and Interval Width¶

Now we will use widgets to see how the width of a confidence interval changes when we:

• Increase or decrease the sample size (n).

• Increase or decrease the confidence level (for example, from 90% to 99%).

Use the sliders below to experiment and watch what happens to the interval.

As you experiment, look for these patterns:

• Larger sample size (n) usually makes the interval narrower (more precise).

• Higher confidence level (for example, 99% instead of 90%) makes the interval wider (more cautious), because we are asking for more “coverage” of the true parameter.

This trade-off between certainty and precision is central to how confidence intervals are used in practice.

Discussion and Reflection¶

Short Answer: In your own words, describe what a 95% confidence interval for a population mean tells you. Why is it incorrect to say that 95% of individual observations fall inside the interval?

Type your answer below.

Replace with your answer.

Run the cell below to check your answers¶

After completing Exercises 1–3, run the next cell. Use grader.check("q1"), grader.check("q2"), grader.check("q3") to check each exercise and grader.check_all() to run all tests at once (recommended before submitting).

📋 Post-Notebook Reflection Form¶

Thank you for completing the notebook! We’d love to hear your thoughts so we can continue improving and creating content that supports your learning.

If your instructor provides a reflection form link, you can add it here, for example:

👉 Click here to fill out the Reflection Form

🧠 Why it matters:¶

Your feedback helps us understand:

• How clear and helpful the notebook was

• What you learned from the experience

• How your views on statistics and data might have changed

• What topics you’d like to see in the future

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Conclusion¶

In this notebook, you:

• Learned how confidence intervals use sample data to estimate unknown population parameters.

• Built confidence intervals for a mean using both t-based formulas and bootstrap methods with numpy and scipy.stats.

• Saw how confidence intervals connect to hypothesis tests and how their width depends on sample size and confidence level.

These tools are central to inferential statistics and will reappear whenever you use data to make claims about a broader population.