Interval Estimation for Population Mean (Sigma Unknown) - T-Distribution

TL;DR

When the population standard deviation (σ) is unknown, especially with a small sample, you use the t-distribution to create an interval estimate for the population mean. This method relies on the sample standard deviation (s) and accounts for the increased uncertainty compared to when σ is known. As your sample size grows, the t-distribution's shape approaches that of the z-distribution.

1. The Mental Model

Imagine you're trying to guess the average height of all students at a university, but you don't know how spread out their heights usually are. Instead, you take a small sample and use its spread to help you estimate the average height for everyone, realizing your estimate will have a bit more wiggle room because you're working with less initial information.

2. The Core Material

Interval estimation is about creating a range where you're confident the true population mean (μ) lies. When you don't know the population standard deviation (σ), and especially when your sample size is small (generally n < 30), you can't use the standard z-distribution. This is where the t-distribution comes in.

Why the t-Distribution?

The t-distribution is used when:
* The population standard deviation (σ) is unknown.
* You use the sample standard deviation (s) instead as an estimate for σ.
* The sample size is small (though it's technically applicable for any sample size when σ is unknown).

Its shape is also bell-shaped and symmetric, similar to the z-distribution. However, the t-distribution is:
* Wider and has heavier tails than the z-distribution. This reflects the increased uncertainty when you're estimating σ from your sample.
* It approaches the z-distribution as your sample size increases. This is because with more data, your sample standard deviation (s) becomes a better estimate of the true population standard deviation (σ).

The General Form of an Interval Estimate

Regardless of whether σ is known or unknown, the general idea for an interval estimate of a population mean is:

Sample Mean ± Margin of Error

When σ is unknown, the interval estimate for μ is specifically:

$\bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}$

Where:
* $\bar{x}$ is the sample mean.
* tα/2 is the t-value from the t-distribution table. This value depends on your desired confidence level and the degrees of freedom (df), which are n - 1 (sample size minus 1).
* s is the sample standard deviation.
* n is the sample size.
* $\frac{s}{\sqrt{n}}$ is the estimated standard error of the mean.

The tα/2 value essentially replaces the zα/2 value used when σ is known.

How Confidence Levels Affect the Interval

The confidence level (like 90%, 95%, 98%) determines how wide your interval estimate will be. A higher confidence level means a wider interval, as you're trying to be more certain that the interval captures the true population mean.

Here’s a diagram illustrating the decision-making process for choosing between a z-distribution or a t-distribution for interval estimation of a population mean:

graph TD
    A["Interval Estimation: Population Mean (μ)"] --> B{"Is Population Standard Deviation (σ) Known?"};
    B -- Yes --> C["Known Case: Use z-distribution"];
    C --> D["Interval Estimate: x̄ ± zα/2 * (σ/√n)"];
    B -- No --> E["Unknown Case: Use t-distribution"];
    E --> F["Use Sample Standard Deviation (s) to estimate σ"];
    F --> G["Interval Estimate: x̄ ± tα/2 * (s/√n)"];

Meaning of Confidence

When you state, "We are 95% confident that the mean rent per month is between $720 and $800,” it means if you were to repeat the sampling and interval estimation process many times, about 95% of the intervals you construct would actually contain the true population mean. It doesn't mean there's a 95% chance that this specific interval contains the population mean; rather, it reflects the reliability of the method over many repetitions.

3. Worked Example

Let's use the Apartment Rents example from your source material, but assume σ is unknown.

A reporter wants a 95% confidence interval for the mean rent of one-bedroom apartments. They collect a sample of 25 apartments (n = 25) and find the following:

• Sample Mean (x̄): $750
• Sample Standard Deviation (s): $80

Steps:

1. Identify Knowns:

   • n = 25
   • x̄ = $750
   • s = $80
   • Confidence Level = 95%
   • Since σ is unknown and n < 30, we use the t-distribution.

2. Determine Degrees of Freedom (df):

   • df = n - 1 = 25 - 1 = 24

3. Find the t-value (tα/2):

   • For a 95% confidence level, α = 1 - 0.95 = 0.05.
   • So, α/2 = 0.025.
   • Look up tα/2 in a t-distribution table with df = 24 and α/2 = 0.025. You'd find t0.025, 24 = 2.064.

4. Calculate the Margin of Error (ME):

   • ME = tα/2 * (s/√n)
   • ME = 2.064 * (80 / √25)
   • ME = 2.064 * (80 / 5)
   • ME = 2.064 * 16
   • ME = 33.024

5. Construct the Confidence Interval:

   • Interval = x̄ ± ME
   • Interval = 750 ± 33.024
   • Lower Limit = 750 - 33.024 = 716.976
   • Upper Limit = 750 + 33.024 = 783.024

Therefore, we are 95% confident that the mean rent per month for one-bedroom apartments within a half-mile of campus is between $716.98 and $783.02.

4. Key Takeaways

• When the population standard deviation (σ) is unknown, you must use the t-distribution for interval estimation of the population mean.
• The t-distribution accounts for the extra uncertainty from using the sample standard deviation (s) as an estimate for σ.
• The key parameters for using the t-distribution are the confidence level and the degrees of freedom (n - 1).
• As the sample size (n) increases, the t-distribution becomes more like the z-distribution.
• A higher confidence level always leads to a wider interval because you're trying to be more certain.

Common Mistakes to Avoid:
* Using the z-distribution when σ is unknown: This is a fundamental error that underestimates the variability.
* Forgetting to subtract 1 for degrees of freedom (df = n - 1): This will lead to an incorrect t-value.
* Confusing sample standard deviation (s) with population standard deviation (σ): Always check which one you have.
* Misinterpreting confidence: It's about the reliability of the method, not the probability of a specific interval.

5. Now Try It

A new coffee shop wants to estimate the average time a customer spends in their store. They take a random sample of 16 customers and find the average time is 12 minutes with a sample standard deviation of 3 minutes.

Provide a 90% confidence interval for the true mean time all customers spend in the coffee shop.

What success looks like: You should be able to state the lower and upper bounds of the confidence interval and interpret what that interval means in plain language.