TDIST

For common statistics, such as the Student s t value, chi-square, and Fisher F, Excel has functions that return the critical value at a given probability and degrees of freedom (e.g., =TINV (0.05,10) for the two-tailed Lvalue at a probability of 95% and 10 degrees of freedom), or which accept a calculated statistic and give the associated probability (e.g., =TDIST( t, 10, 2 ) for 10 degrees of freedom and two tails). Table 2.3 gives common statistics calculated in the course of laboratory quality control. 

There are three ways to perform a t test in Excel. First, a t value can be calculated from the appropriate equation and then the probability calculated from =TDIST (t, df, tails ), where df = degrees of freedom and tails = 2 or... 

The probability of a difference this large occurring because of only random errors can be obtained from the Excel function TDIST(x,deg freedom,tails), where x is the test value of /(4.375), deg freedom is 3 for our case, and tails = 2. The result is TDIST(4.375,3,2) = 0.022. Hence, it is only 2.2% probable to get a value this large because of random error. The critical value of t for a given confidence level can be obtained in Excel from TINV(probability,deg freedom). In our case, TINV(0.05,3) = 3.1825. 

The critical value of t at the 95% confidence level for 10 — 2 = 8 degrees of freedom is 2.31. Since 1.771 < 2.31, we accept the null hypothesis at the 95% confidence level and conclude that there is no difference in the alcohol content of the wines. The probability of getting a t value of 1.771 can be calculated using the Excel function TD1ST() and is TDIST( 1.771,8,2) = 0.11. Thus, there is more than a 10% chance that we could get a value this large due to random error. 

Spreadsheet Summary In the first exercise in Chapter 3 of Applications of Microsoft Excel in Analytical Chemistry, we use Excel to perform the t test for comparing two means assuming equal variances of the two data sets. We first manually calculate the value of t and compare it with the critical value obtained from Excel s function TINV(). We obtain the probability from Excel s TDIST() function. Then, we use Excel s built-in function TTEST() for the same test. Finally, we employ Excel s Analysis ToolPak to automate the t test with equal variances. 

Calculate the probability of a Student t-value =TDIST(f, df, tails) df = degrees of freedom tails = 1 (one-tailed) or 2 (two-tailed)... 

TINV(p,df)) are two tailed. However, when the probability of finding a Student t (T) greater than a particular / is to be calculated using TDIST, whether the one-tailed or two-tailed values are to be returned must be specified (=TDIST(f, df, tails)). 

This can simply be done in Excel using the function TDIST with syntax TDIST(f, df, tails). Therefore in a blank cell of a spreadsheet you would input =TDIST(2.23, 10, 2). 

If =TDIST(2.23, 10, 1) were typed into a cell in the spreadsheet then the output would be 0.024921 (i.e., 0.025) as this refers to a one-tail probability. 

Calculate the probability of this t- value using TDIST(f, n — 1, tails), which gives p = 0.0103. 

Calculate the associated probability for the /-statistic using TDIST(f, df, tails) which gives the answer p = 0.017828 (see spreadsheet 3.6). 

From the t-value calculate a probability using =TDIST (t, n — 1, tails), where n is the number of pairs, which for this set of data is =TDIST(2.6905, 8, 2), see spreadsheet 3.7. p = 0.0275 means if we were to reject the null hypothesis we would do so knowing we would make an error in 2.75% of repeated experiments. As this is less than the usual 5% given... 

Figure 13.5. Trendofdistillatefluxes. Tfeed =60 C Tdist =21-23°C cas =94p.gL Vfeed =0.028ms =0.052ms (adapted from Pal and Manna, 2010).

Figure 13.8. Variation of the permeate flux with fluoride concentrations, ufeed =0.52ms Ifted =50°C Vdist = 0.15 m s" Tdist = 20°C (adapted from Hou et al., 2010).

Figure 13.12. ElFect of die uranium concentration on the permeate flux for the PTFE membrane. Tfeed = 80°C Tdist = 20°C gfeed = fidist =4 L min (adapted from Yarlagadda et al., 2011).

As DCMD (Flat sheet) PVDF Rejection = 100% 7 = 95kgm-2h-i 02feed = 120Lh-i 2dist = 150Lh-i Tfeed = 60°C Tdist = 20-22X cas=396 ixgL ) Manna et al. (2010)... 

---