LESSON 14: STUDENT t DISTRIBUTION Outline • Student t distribution • Table • Excel • Example
STUDENT t DISTRIBUTION FOR SAMPLETHE POPULATION VARIANCE IS UNKNOWN • If the population variance, is not known, we cannot compute the z-statistic as • However, we may compute a similar statistic, the t-statistic, that uses the sample standard deviation s in place of the population standard deviation :
STUDENT t DISTRIBUTION FOR SAMPLETHE POPULATION VARIANCE IS UNKNOWN • If the sampled population is normally distributed, the t-statistic follows what is called Student t distribution. • Student t-distribution is similar to the normal distribution. The Student t-distribution is • symmetrical about zero • mound-shaped, whereas the normal distribution is bell-shaped • more spread out than the normal distribution. • The difference between t-distribution and normal distribution depends on degrees of freedom, d.f. = n-1. For small d.f., the difference is more. For large d.f., the t-distribution approaches the normal distribution. (See next)
STUDENT t DISTRIBUTION: TABLE • Notation: • tαis that value of t for which the area to its right under the Student t-curve equals α. So, • tα,dfis that value of t for which the area to its right under the Student t-curve for degrees of freedom=df equals α. • The value of tα,df is obtained from Table G on p. 541, Appendix A. • The table provides t-values for given areas. However, the table does not give areas for all t-values. Excel may be used to find areas for all t-values. See next.
STUDENT t DISTRIBUTION: EXCEL • Some of the Excel functions for Student t-distribution are: • TDIST(t, df, number of tails): Given t, df, and number of tails, finds area in the tail(s) • For example, TDIST(2,60,1) = 0.025 • This means that for t=2 and for degrees of freedom = 60, the area to the right of t=2 is 0.025. • Also, for t=-2 and degrees of freedom 60, the area to the left of t=-2 is 0.025. • TINV(Tail area, df ): Assumes two-tails. Given area in two tails and df finds t. To get t for one tail, multiply the area by 2.
STUDENT t DISTRIBUTION: EXCEL • For example, TINV(0.1,35) 1.60. • This means that for area in two tails = 0.10 and for degrees of freedom =35, t=1.60 • This also means that for area in the right tail = 0.10/2 = 0.05 and for degrees of freedom = 35, t=1.60 • And for area in the left tail = 0.10/2 = 0.05 and for degrees of freedom = 35, t=-1.60. • So, to get tα,df use the command TINV(2α, df). Note that the area is multiplied by 2.
STUDENT t DISTRIBUTION: EXAMPLE Example 1: Using t distribution, find critical values for the following tail areas: • α=.05, d.f. = 9 • α=0.10, d.f.=5
STUDENT t DISTRIBUTION: EXAMPLE Example 2: Find the probability that a compute t from a sample of size 20 will fall: • Above 2.093 • Below 1.729 • Between 1.729 and 2.093
STUDENT t DISTRIBUTION: EXAMPLE Example 3: A chemical engineer has the following results for the active ingredient yields from 16 pilot batches processed under a retorting procedure: Determine the approximate probability for getting a result this rare or rarer if the true mean yield is 30.5 grams/liter.
READING AND EXERCISES Lesson 14 Reading: Section 8-3, pp. 277-280 Exercises: 9-23, 9-25, 9-28