6.3 Other Levels of Confidence

Dr. Lauren Perry

Goals

Use the normal distribution applet to find critical values.
Find and interpret confidence intervals for a mean when (A) the population is normal and (B) $\sigma$ is known.

The 95% confidence interval is common in research, but there’s nothing inherently special about it.

You could calculate a 90%, a 99%, or even something like a 43.8% confidence interval.
These numbers are called the confidence level.
- They represent the proportion of times that the parameter will fall in the interval (if we took many samples).

Confidence Intervals

The 100(1- $\alpha$ )% confidence interval for $\mu$ is given by $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ where $z_{\alpha/2}$ is the z-score associated with the $[1-(\alpha/2)]$ th percentile of the standard normal distribution.

Critical Values

The value $z_{\alpha/2}$ is called the critical value (“c.v.” on the plot, below).

Common Critical Values

Confidence Level	$\alpha$	Critical Value, $z_{\alpha/2}$
90%	0.10	1.645
95%	0.05	1.96
98%	0.02	2.326
99%	0.01	2.575

Example

Prior experience with SAT scores in the CSU system suggests that SAT scores are well-approximated by a normal distribution with standard deviation known to be 50. Suppose we have a random sample of 50 Sac State students with (sample) mean SAT score 1112.

Calculate a 98% confidence interval.
Interpret the interval in the context of the problem.

Checkpoint

Calculate a 90% confidence interval.
Interpret each interval in the context of the problem. Comment on how the intervals change as you change the confidence level.

Breaking Down a Confidence Interval

$\left(\bar{x}- z_{\alpha/2}\frac{\sigma}{\sqrt{n}}, \quad \bar{x} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right)$ The key values are

$\bar{x}$ , the sample mean
$\sigma$ , the population standard deviation
$n$ , the sample size
$z_{\alpha/2}$ , the critical value $P(Z > z_{\alpha/2}) = \frac{\alpha}{2}$

Breaking Down a Confidence Interval

$\left(\bar{x}- z_{\alpha/2}\frac{\sigma}{\sqrt{n}}, \quad \bar{x} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right)$

The value of interest is $\mu$ , the (unknown) population mean.
The confidence interval gives us a reasonable range of values for $\mu$ .

In addition, the formula includes

The standard error, $\frac{\sigma}{\sqrt{n}}$
The margin of error, $z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$

Confidence Level

$\left(\bar{x}- z_{\alpha/2}\frac{\sigma}{\sqrt{n}}, \quad \bar{x} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right)$

If we can be 99% confident (or even higher), why do we tend to “settle” for 95%??

What will higher levels of confidence do to this interval?

Homework Problems

Section 6.3 Exercises 1 and 2