5.4.1 - Construct and Interpret the CI

5.4.1 - Construct and Interpret the CI

Constructing a Confidence Interval for the Population Mean

To construct a confidence interval for a population mean, we're going to apply the same three steps as with the population proportion, but first, let's look at the two possible cases.

Case 1: $\sigma$ is known

In the previous lesson, we learned that if the population is normal with mean $\mu$ and standard deviation, $\sigma$, then the distribution of the sample mean will be Normal with mean $\mu$ and standard error $\frac{\sigma}{\sqrt{n}}$.

Following the similar idea to developing the confidence interval for $p$, the $(1-\alpha)$100% confidence interval for the population mean $\mu$ is...

$P\left(\left|\dfrac{\bar{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}\right|\le z_{\alpha/2}\right)=1-\alpha$

A little bit of algebra will lead you to...

$P\left(\bar{x}-z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\le \mu\le \bar{x}+z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\right)=1-\alpha$

In other words, the $(1-\alpha)$100% confidence interval for $\mu$ is:

$\bar{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$

Notice for this case, the only condition we need is the population distribution to be normal.

Note!

The case where $\sigma$ is known is unrealistic. We explain it here briefly because it reinforces what we have previously learned. We do not present examples in this case.

Case 2: $\sigma$ is unknown

When the population is normal or when the sample size is large then,

$Z=\dfrac{\bar{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}$

where Z has a standard Normal distribution.

Usually, we don't know $\sigma$, so what can we do?

Recall that if X comes from a normal distribution with mean, $\mu$, and variance, $\sigma^2$, or if $n\ge 30$, then the sampling distribution will be approximately normal with mean $\mu$ and standard error, $SE(\bar{X})=\frac{\sigma}{\sqrt{n}}$

One way to estimate $\sigma$ is by $s$, the standard deviation of the sample, and replace $\sigma$ by $s$ in the above Z-equation. However, this new quotient no longer has a Z-distribution. Instead it has a t-distribution. We call the following a 'studentized' version of $\bar{X}$:

$t=\dfrac{\bar{X}-\mu}{\dfrac{s}{\sqrt{n}}}$

Constructing the Confidence Interval

1. CHECK THE CONDITIONS

One of the following conditions need to be satisfied:

1. If the sample comes from a Normal distribution, then the sample mean will also be normal. In this case, $\dfrac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}$ will follow a $t$-distribution with $n-1$ degrees of freedom.
2. If the sample does not come from a normal distribution but the sample size is large ($n\ge 30$), we can apply the Central Limit Theorem and state that $\bar{X}$ is approximately normal. Therefore, $\dfrac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}$ will follow a $t$-distribution with $n-1$ degrees of freedom.
2. CONSTRUCT THE GENERAL FORM

$(1-\alpha)$100% Confidence Interval for the Population Mean, $\mu$
$\bar{x}\pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}$

where the t-distribution has $df = n - 1$. This interval is also known as the one-sample t-interval for the population mean.

3. INTERPRET THE CONFIDENCE INTERVAL

We are $(1-\alpha)100\%$ confident that the population mean, $\mu$, is between $\bar{x}-t_{\alpha/2}\frac{s}{\sqrt{n}}$ and $\bar{x}+t_{\alpha/2}\frac{s}{\sqrt{n}}$.

What if the conditions are not met?

What will you do if you cannot use the t-interval? What do we do when the above conditions are not satisfied?

1. If you do not know if the distribution comes from a normally distributed population and the sample size is small (i.e $n<30$), you can use the Normal Probability Plot to check if the data come from a normal distribution.
2. You may want to consider what is known as nonparametric statistical methods. A procedure such as the one-sample Wilcoxon procedure. Lesson 11 introduces nonparametric statistical methods.

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