# Chi-Squared confidence intervals

### Chi-Squared confidence intervals

#### Lessons

To estimate a population variance a Chi-Squared distribution is used,
• Chi-Squared: $X^2=\frac{(n-1)s^2}{\sigma ^2}$
$n$: sample size
$s$: sample standard deviation
$\sigma$: population standard deviation
$(n-1)$: is also called "degrees of freedom"
• Chi-Square table gives critical value area to the right

The Confidence interval for the variance is given by:
$\frac{(n-1)s^2}{X_R^2}$ < $\sigma ^2$ < $\frac{(n-1)s^2}{X_L^2}$
• Introduction
What are Chi-Squared Confidence Intervals?

• 1.
Determining Degrees of Freedom
How many degrees of freedom does a sample of size,
a)
7 have?

b)
20 have?

• 2.
Determining the Critical Value for a Chi-Square Distribution $(X_R^2$ and $X_L^2)$
If a Chi-Squared distribution has 8 degrees of freedom find $X_R^2$ and $X_L^2$, with a
a)
95% confidence level

b)
99% confidence level

• 3.
Determining the Confidence Interval for Variance
Road and racing bicycles have an average wheel diameter of 622mm. From a sample of 15 bicycles it was found that the wheel diameters have a variance of 10mm. With a 90% confidence level give a range where the variance of all road and racing bicycle wheels lie.

• 4.
Determining the Confidence Interval for Standard Deviation
A Soda-pop company "Jim's Old Fashion Soda" is designing their bottling machine. After making 41 bottles they find that their bottles have an average of 335mL of liquid with a standard deviation of 3mL. With a 99% confidence level what is the range of standard deviation that this machine will output per bottle?