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Student t distribution Table Excel Example

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STUDENT t DISTRIBUTION FOR SAMPLE THE POPULATION VARIANCE IS UNKNOWN • If the population variance, σ is not known, we cannot compute the z-statistic as x−µ z= σ/ n • However, we may compute a similar statistic, the t-statistic, that uses the sample standard deviation s in place of the population standard deviation σ: X −µ t= s/ n

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STUDENT t DISTRIBUTION FOR SAMPLE THE POPULATION VARIANCE IS UNKNOWN • If the sampled population is normally distributed, the tstatistic 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 bellshaped – 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 tdistribution approaches the normal distribution. (See next) 3

d.f.=1

4

2

d.f.=2

5

d.f.=3

6

3

d.f.=4

7

d.f.=5

8

4

d.f.=30

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STUDENT t DISTRIBUTION: TABLE • Notation: – t a is that value of t for which the area to its right under the Student t-curve equals a. So,α = P(t > t α ). – t a,df is that value of t for which the area to its right under the Student t-curve for degrees of freedom=df equals a. • The value of t a ,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. 10

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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. 11

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 t a,df use the command TINV(2a, df). Note that the area is multiplied by 2. 12

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STUDENT t DISTRIBUTION: EXAMPLE Example 1: Using t distribution, find critical values for the following tail areas: a. a=.05, d.f. = 9

b.

a=0.10, d.f.=5

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STUDENT t DISTRIBUTION: EXAMPLE Example 2: Find the probability that a compute t from a sample of size 20 will fall: a. Above 2.093

b.

Below 1.729

c.

Between 1.729 and 2.093

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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: X = 32 grams/lite r, s = 3 Determine the approximate probability for getting a result this rare or rarer if the true mean yield is 30.5 grams/liter.

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READING AND EXERCISES Lesson 14 Reading: Section 8-3, pp. 277-280 Exercises: 9-23, 9-25, 9-28

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