Sensory Discrimination - Essay Example

Sensory Discrimination Summary This is a nine-page paper describing an experiment to test the sensory discrimination of a friend. The purpose of this study is to investigate if a subject has sensory discrimination on the test products, namely Dominos Meatzza Feast and Pizza Hut Meat Lover’s pizza. The researcher intends to determine if there is any significant association, which is also referred to as contingency, between our two classifications. It relies on seven sources and is in APA format. Experiment to Test the Sensory Discrimination of a Friend Purpose of the Study The purpose of this study is to investigate if a subject has sensory discrimination on the test products namely Dominos Meatzza Feast and Pizza Hut Meat Lover’s pizza. The researcher wants to determine if there is any significant association, which is also referred to as contingency between our two classifications. The test criterion can be that Steve is able to guess the type of Pizza presented to him. In the experiment, the subject was asked if he could identify if the pizza he tastes was a Dominos Meatzza Feast or a Pizza Hut Meat Lover’s pizza. The number of correct guesses was recorded and entered in a table alongside the number of incorrect guesses. The experiment was conducted over a period of six days with a total of three trials per day. In the null hypothesis, we take an opposing position speculating that Steve has no sensory discrimination. Statistical hypothesis determines whether or not there is a difference when the null hypothesis is true. The p-value gives the probability of obtaining results of a trial when the null hypothesis is true. (Weaver, 2008) Hypothesis Let p1 be the proportion of the characteristic of interest in population 1 and p2 be the proportion of the characteristic of interest in population 2. (A) Two-sided: H0: p1 = p2 vs. H1: p1 = p2 (B) Lower one-sided: H0: p1 = p2 vs. H1: p1 < p2 (C) Upper one-sided: H0: p1 = p2 vs. H1: p1 > p2 For the three alternatives, the decided rule is that we reject the null hypothesis when the p-value is less than or equal to alpha. (Speed, 2000) The H values are defined as follows; H0: Steve has no sensory discrimination H1; Steve has sensory discrimination Method In this experiment, we utilize simple counts such as how many pieces of pizza were classified by the subject. The results were summarized in a 2 by 2 contingency table as shown below. The four numbers are a, b, c, and d, which are entered in the cells of the table according to the four categories. The total number of pieces of pizza which the subject can classify as Dominos Meatzza Feast or Pizza Hut Meat Lover’s pizza is: n= a + b + c + d From the table below we see that a total of a + c pieces of pizza were guessed correctly while b + d pieces of pizza were guessed incorrectly. If the subject cannot distinguish between Dominos Meatzza Feast and Pizza Hut Meat Lover’s pizza then a and c should be small. Conversely, if the subject can tell the difference between the two pizzas, we expect a and c to be about the same. (Weaver, 2008)   Guessed correctly Guessed incorrectly Total Dominos Meatzza Feast a b a + b Pizza Hut Meat Lover’s pizza c d c + d Totals a + c b + d a + b + c + d (=n) According to Fisher the probability of obtaining the values base above is given by the hypergeometric distribution below: (Deborah, 2000) The p-value is calculated in such a way that the margins of the table are fixed and, therefore, the respondent will give guesses with correct numbers in each category of the table. This leads to a null hypothesis of independence to a hypergeometric distribution of the entries in the cells of the contingency table. When using the chi-square in the analysis, the approximation is not enough when sample sizes are small or if the data is very unequally distributed in the cells of the table. Therefore the cell counts have been predicted to be low. (Agresti, 2001) In this experiment, 18 pieces of pizza were prepared--9 pieces of Dominos Meatzza Feast and 9 pieces of Pizza Hut Meat Lover’s pizza. The pieces of pizza were presented to the respondent in a random order. The respondent’s task is to identify the 9 pieces of Dominos Meatzza Feast and the 9 pieces of Pizza Hut Meat Lover’s pizza, and he is informed of the design. Participants The participant in this study was Steve David. He claims to be able to tell the difference between Dominos Meatzza Feast and Pizza Hut Meat Lover’s pizza since he runs a pizza store, and thus qualifies as an expert on the subject. Experimental Design The Fisher test is used with respect to the contingency table results. In this case, the test assesses what the expected frequencies will be if the null hypothesis is true. If there is no difference between Dominos Meatzza Feast and Pizza Hut Meat Lover’s pizza guesses, one would expect that Dominos Meatzza Feast would have been guessed correctly six times. According to Siegel this can be calculated as (9 * 12) / 18 = 108 / 18 = 6) We can formulate our null hypothesis such that we speculate that the respondent has no sensory discrimination. By randomization of the order of the pieces of pizza, this makes the 9 pieces of pizza chosen by the respondent as Dominos Meatzza Feast equally likely to be any of the 18 pieces served. There are (18 combinations 9) ways of classifying 9 of the pieces as Dominos Meatzza Feast. If the respondent has no ability to discriminate between Domino Meatzza Feast and Pizza Hut Meat Lover's pizza, then by randomization each of the (18 combinations 9) ways are possible. (Siegel, 1988) Independent trials were conducted where the respondent was provided with one piece of either Dominos Meatzza Feast or Pizza Hut Meat Lover's pizza and asked to identify which one it was. The subject was given the pieces of pizza in a random order without replacement. The trials were independent since after each trial the respondent was asked to rinse so that that one particular trial does not affect the next trial. The numbers of trials conducted were 18. The binomial distribution was kept in mind since the trials were either a success or a failure as in this particular case the subject guessed correctly or incorrectly. In this experiment, there were no unusual occurrences. (Agresti, 2000) The generally accepted way of calculating the P value for Fisher's exact test has been proved conservative. That means that requires more evidence than is necessary to reject a false null hypothesis. The Fisher Exact test is conducted on a 2 by 2 table. When one is given the marginal frequencies, the test assumes that in the population the two factors (Dominos Meatzza Feast and Pizza Hut Meat Lover’s pizza) have no relationship. For small n and this probability can be determined exactly by counting all possible tables that can be formed based on the marginal frequencies. Chi-square test assumes that each cell has an expected frequency greater than five but in the Fisher's exact test does not have this assumption and can be used in spite of how small the expected frequency is. (Pierce, 1999)Fisher’s exact test is always appropriate, even when the sample size is large (Stokes, 1995), and that Fisher's exact test is considered a non-parametric test (Walker, 1997). Data collection Collection of data was from a random respondent who has been selected with known probabilities of selection from among all units in the whole population universe. In this case, there are no assumptions about the representation which is needed for estimating totals or averages for the target universe. There are well-known techniques for determining the precision of these estimates were used. (Deborah, 2001) Sample size What is required is a sample which is significantly large enough to provide estimates of the balances and other parameters of interest in the experiment, and which is reliable enough to meet the requirements of the person conducting the experiment. This means that in order to be able to determine the appropriate sample size, we need to know the level of precision required by the one conducting the experiment and the precision that can be obtained from the different samples. (Fisher, 1922). The sample size was 18 pieces, all of which were uniform and were of the same size. Data Analysis Fisher’s exact test caters for all the possible cell combinations, which would still have the marginal frequencies. The test is exact since it uses the exact hypergeometric distribution instead of the most commonly used chi-square distribution to determine the p-value. The p-value that was calculated using Fisher’s exact test is 0.1312. Therefore, we fail to reject the null hypothesis of equal proportions at α = 0.05 level. We conclude that the respondent has no sensory discrimination. (Siegel, 1988)Cross-tabulations allow us to identify relationships between the cross-tabulated variables. Results After the eighteen trials were completed Steve was able to identify correctly 8 pieces as Dominos MeatzzaFeast and four pieces of pizza as Pizza Hut Meat Lover’s pizza. The subject also identified 1 piece of Dominos Meatzza Feast incorrectly and five pieces of Pizza Hut Meat Lover’s pizza incorrectly. This is shown in Table 1.1.All the two items namely, Dominos meatzza and pizza hut have the same probability of being guessed.It’s appropriate to use Fisher’s exact test, in particular when dealing with small counts.The number of possible tables with the above marginal totals is 6 and the tables are illustrated as Tables 1.2-1.7. In Table 1.2, 9 pieces were guessed correctly to be Dominos Meatzza, 3 pieces were guessed incorrectly to be Pizza hut while 6 pieces are guessed incorrectly to be Pizza hut. Using the formula for hyper-geometric distribution,we determine that the probability of the Table 1.2 is given by: (9!*9!*12!*6!)/(18!*9!*0!*3!*6!) =0.004524887 * In the Table 1.3 the number of correct guesses includes 7 pieces of Dominos Meatzza and 5 pieces of Pizza Hut Meat Lover’s pizza. The number of pieces guessed incorrectly is 2 pieces of Dominos Meatzza and 4 pieces of Pizza Hut Meat Lover’s pizza. Using the formula for hyper-geometric distribution we determine that the probability of the table 1.3 is given by: (9!*9!*12!*6!)/ (18!*7!*2!*5!*4!) = 0.244343891 In the Table 1.4 the number of correct guessesincludea 6 Dominos Meatzza and 6 pieces of Pizza Hut Meat Lover’s Pizza. The number of incorrect guesses is 3 pieces of Dominos Meatzza and 3 pieces of Pizza Hut Meat Lover’s Pizza. Using the formula for hyper-geometric distribution we determine that the probability of the table 1.4 is given by: (9!*9!*12!*6!) / (18!*6!*3!*6!*3!) = 0.380090498 In the Table 1.5 the number of piece guessed correctly is 5 pieces of Dominos Meatzza and 7 pieces of Pizza Hut Meat Lover’s Pizza. The number of pieces guessed incorrectly includes 4 pieces of Dominos Meatzza and 2 pieces of Pizza Hut Meat Lover’s Pizza. Using the formula for hyper-geometric distribution we determine that the probability of Table 1.5 is given by: (9!*9!*12!*6!) /(18!*5!*4!*7!*2!) = 0.24434 3891 From Table 1.6 the number of pieces guessed correctly is 4 pieces of Dominos Meatzza and 8 pieces of Pizza Hut Meat Lover’s pizza. The number of pieces guessed incorrectly is 5 pieces of Dominos Meatzza and 1 piece of Pizza Hut Meat Lover’s Pizza. Using the formula for hyper-geometric distribution we determine that the probability of the table 1.6 is given by: (9!*9!*12!*6!) / (18!*4!*5!*8!*1!) =0.061085973 * In Table 1.7, 3 pieces are guessed correctly to be Dominos Meatzza and 9 pieces are guessed correctly to be Pizza Hut Meat Lover’s pizza. Meanwhile, 6 pieces are guessed incorrectly to be Dominos Meatzza. Using the formula for hyper-geometric distribution we determine that the probability of Table 1.7 is given by: (9!*9!*12!*6!) / (18!*3!*6!*9!*0!) = 0.004524887 * To determine Fisher’s exact test results we look at the tables with probabilities less than or equal to the probability of the observed results (0.061085972). These probabilities are mark with an*. Using the addition rule for mutually exclusive events we can add the probabilities from the three tables, namely Table 1.2, Table 1.6 and Table 1.7. Adding the probabilities to the probability of the observed results we get the p-value for the test.The exclamation mark denotes factorial function and thusindicates successive multiplication by cardinal numbers in descending series. For example, 4! means 4 x 3 x 2 x 1. (Siegel, 1988) Conclusion Since the exact conditional probability of sensory discrimination where p = 0.1312 which is greater than 0.05, then by the decision rule we accept the null hypothesis that the subject has no sensory discrimination. The use of random samples and avoiding of using united data confirms the validity of the Fisher Exact test, and the data produced through interaction with the respondent provides a proper amount for analysis. Overall, the null hypothesis indeed accepted due to the support given by the data and results. Appendix Table 1.1 = [8, 1, 4, 5] Left : p-value = 0.9954751131221721 Right: p-value = 0.06561085972850711 2-Tail: p-value = 0.13122171945 Steve guessed correctly Steve guessed incorrectly total Dominos 8 1 9 Pizza hut 4 5 9 Total 12 6 18 _____________________________________________________________________________ Table 1.2: Guessed correctly Guessed incorrectly Total Dominos 9 0 9 Pizza Hut 3 6 9 Total 12 6 18 Table 1.3: Guessed correctly Guessed incorrectly Total Dominos 7 2 9 Pizza Hut 5 4 9 Total 12 6 18 Table 1.4: Guessed correctly Guessed incorrectly Total Dominos 6 3 9 Pizza Hut 6 3 9 Total 12 6 18 Table 1.5: Guessed correctly Guessed incorrectly Total Dominos 5 4 9 Pizza Hut 7 2 9 Total 12 6 18 Table 1.6: Guessed correctly Guessed incorrectly Total Dominos 4 5 9 Pizza Hut 8 1 9 Total 12 6 18 Table 1.7: Guessed correctly Guessed incorrectly Total Dominos 3 6 9 Pizza Hut 9 0 9 Total 12 6 18 References Agresti, A. (2001). Exact inference for categorical data: recent advances and continuing Controversies. London: Longhorn Publishers Deborah .A, Speed .T (2000) Stat labs: mathematical statistics through applications: New York: Macmillan Fisher, R. A. (1922). On the interpretation of χ2 from contingency tables, and the calculation ofP. Journal of the Royal Statistical Society 85 (1): 87–94. Keith M. B, S. ( 2007) When to use fisher test. Retrieved on 13 September 2011 from http://www.minitab.com/uploadedFiles/Shared_Resources/Documents/Articles/fisher_exact_test.pdf  Pierce, D.A. & Peters D. (1999).Improving on exact test by approximate conditioning.New York: Cengage Learning Siegel, S. and Castellan, Jr. (1988), Nonparametric Statistics for the Behavioral Sciences.McGraw–Hill, New York. Weaver.B (2008) Fisher Exact test. Retrieved on 13 September 2011 fromhttp://www.langsrud.com/fisher.htm Read More