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

Objective Perform a T-Test

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

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)

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

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

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

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

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

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?

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.

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

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.

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?

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

Example We obtain the following sample statistics: Use these results to test the claim that men have a mean weight greater than 166.3 lb. Use a 0.05 significance level. Use the traditional method.

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 166.3 lb. Use a 0.05 significance level. 1.69 1.50 It’s NOT in the dark – fail to reject that Ho!