On the previous page, we used a confidence interval to estimate the population mean, µ. For this confidence interval, we had to supply a guess for the population standard deviation, σ, based on previous studies. It may have occurred to you that if we do not know µ, it is unlikely that we know σ. So we now take a different approach. We estimate σ using the sample standard deviation, s.

This is the same type of adjustment we used in Inference for One Proportion when we had to adjust our model of the sampling distribution. The standard error of the sampling distribution is [latex]\sqrt{p(1-p)/n}[/latex]. (If we knew [latex]p[/latex], then we wouldn’t need to build a confidence interval.) We approximate [latex]p[/latex] by the sample proportion, [latex]\stackrel{ˆ}{p}[/latex].

Our process for adjusting the confidence interval for estimating µ is similar. We use the sample standard deviation, s, to estimate σ. The standard error for the sampling distribution [latex]σ\text{}/\sqrt{n}[/latex] becomes [latex]s\text{}/\sqrt{n}[/latex]. So we adjust the margin of error in the confidence interval formula, but this adjustment is not as straightforward as our work with proportions.

This estimate for σ introduces more uncertainty in the process. The problem is worse with smaller samples because the sample standard deviations vary more. For small samples, s is a worse approximation for σ. Unfortunately, this approximation makes the normal model a bad fit and inappropriate for determining critical values. We instead use what is called a t-model for this purpose.

Introduction to the T-Model

Here is the formula for the T-score. We also include the z-score for comparison. The formulas are very similar.

[latex]Z\text{}=\text{}\frac{\stackrel{¯}{x}-μ}{\frac{σ}{\sqrt{n}}}\text{ }T\text{}=\text{}\frac{\stackrel{¯}{x}-μ}{\frac{s}{\sqrt{n}}}[/latex]

The distribution of z-scores is the standard normal curve, with mean of 0 and standard deviation of 1. The distribution of T-scores depends on the sample size, n. There is a different T-model for every n. So the T-model is a family of curves.

Instead of referring to n to specify which T-model to use, we refer to the degrees of freedom, or df for short. For Topics 10.2 and 10.3, the number of degrees of freedom is 1 less than the sample size. That is, df = n – 1.

In summary, a normal model is defined by its mean and standard deviation. A T-model is a family of curves defined by the degrees of freedom.

Let’s take a look at a few T-model curves (for various df) to see how they compare to the normal model.

We can see from the picture that as df grows, the T-model gets closer to the standard normal model.

Similarities between T-model and standard normal model:

• Symmetric with a central peak, bell-shaped.
• Centered at 0.
• The larger the degrees of freedom, the closer the T-model is to the standard normal model.

Difference between T-model and standard normal model:

• The T-model has more spread than the standard normal model.
• The T-model has more probability in the tails and less in the center than the standard normal model. We can see this in the fatter tails and lower central peak of the T-model.

When is a T-model a good fit for the sampling distribution of sample means?

Check these conditions before using the T-model:

• Use the T-model if σ (the population standard deviation) is unknown. If σ is known, then use the normal model instead of the T-model.
• Use the T-model if variable values are normally distributed in the population. If this is not true, then make sure the sample size is large (more than 30).

Confidence intervals at the 95% confidence level are common in practice. But 95% is not the only confidence level we use. Particularly in situations that involve safety issues, such as the previous example, people often prefer to estimate population means with 99% confidence intervals. Let’s do some exploration with technology to see how changes in the confidence level affect the confidence interval.

Click here to open this simulation in its own window.