1. Hypothesis Tests Small Sample Mean
The T-Test
Population St Dev UNKNOWN

2. Objective
Perform a T-Test

3. Relevance
Be able to use sample statistics to test population parameters.

4. Testing a Claim About a Mean – Population Standard Deviation Unknown
The Z-Test is how we tested a claim when we knew the population standard deviation.
The Z-Test is based on the unrealistic assumption that we know the population standard deviation.
The T-Test is a way to test the claim about a mean when nothing is known about the population standard deviation. (more realistic)

5. Properties of the T Distribution
Different for different sample sizes
Same basic shape as the standard normal distribution but it is wider to reflect the greater variability expected when using s.
Has a mean of t = 0.
The standard deviation varies with the sample size and is greater than 1 (unlike a z)
As the sample size n gets larger, the t distribution gets closer to standard normal

6. Test Statistic and Critical Value (s) for a T-Test
Remember: T-chart is based on degrees of freedom (n – 1).

7. CV Found in Chart or Using GDC
Chart Practice
Find the critical value (CV) for a sample of size 17, at ,
right tailed.
Look Up Values
CV Found in Chart or Using GDC
Use the row that says amount of alpha in One Tail.
df = 16

8. CV Found in Chart or Using GDC
Chart Practice
Find the critical value (CV) for a sample of size 23, at ,
left tailed.
Look Up Values
CV Found in Chart or Using GDC
Use the row that says amount of alpha in One Tail.
df = 22

9. CV Found in Chart or Using GDC
Chart Practice
Find the critical value (CV) for a sample of size 19, at ,
two tailed.
Look Up Values
CV Found in Chart or Using GDC
Use the row that says amount of alpha in Two Tails.
df = 18

10. T-Test Example
A machine is designed to fill jars with 16 oz. of coffee. A consumer suspects that the machine is not filling the jars completely. A sample of 8 jars has a mean of 15.6 oz. and a st. dev. of 0.3 oz. Is there enough evidence to support the claim at
= 0.10?

11. Example – Traditional Method
A machine is designed to fill jars with 16 oz. of coffee. A consumer suspects that the machine is not filling the jars completely. A sample of 8 jars has a mean of 15.6 oz. and a st. dev. of 0.3 oz. Is there enough evidence to support the claim at = 0.10?
-1.42
-3.77
It’s in the dark - you better reject that Ho!
Support Claim.

12. Example
A job placement director claims that the average starting salary for nurses is $24,000. A sample of 10 nurses has a mean of $23,450 and a st. dev. of $400. Is there enough evidence to reject the director’s claim at = .05?
Values:
n = 10
df = 9
=.05

13. Example – Traditional Method
A job placement director claims that the average starting salary for nurses is $24,000. A sample of 10 nurses has a mean of $23,450 and a st. dev. of $400. Is there enough evidence to reject the director’s claim at a significance level of 0.05?
-4.35
-2.26
2.26
It’s in the dark - you better reject that Ho!
DNS Claim.

14. Example
A doctor claims that a jogger’s maximum volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 43.6 ml per kg. and a st. dev. of 6. If the average of all adults is 36.7 ml per kg, is there enough evidence to support the claim at =.01?

15. Answer 2.62 4.45 It’s in the dark - you better reject that Ho!
Support Claim.

16. Example We obtain the following sample statistics:
Use these results to test the claim that men have a mean weight greater than lb.
Use a 0.05 significance level.
Use the traditional method.

17. Example – Traditional Method
Use the following sample statistics of the weights of men to test the claim that men have a mean weight greater than lb. Use a 0.05 significance level.
1.69
1.50
It’s NOT in the dark – fail to reject that Ho!