LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2

Transcription

1 LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2 Data Analysis: The mean egg masses (g) of the two different types of eggs may be exactly the same, in which case you may be tempted to accept your null hypothesis. It is more likely, however, that the two sample means will differ by some amount. Are the means different enough to enable you to reject your null hypothesis? Remember, the null hypothesis is there is no difference in the average mass in grams between the two different types of eggs. Your sample means are really only estimates of the true mean of egg masses (g) of the two different types of eggs that you chose to compare. Although you have a total of 12 observations, your samples are still small relative to the total number of eggs sold in stores across the state. In order to test your hypothesis, you must have some basis for deciding whether or not the difference between the two sample means could have arisen simply by chance when, in fact, there is no real difference in the average egg mass (g) of the two different types. In other words, before you can compare them, you need to know how accurately your sample means represent the true means. The accuracy of any sample mean is related to: 1) the amount of variation in the data that were collected; and 2) the number of observations (n). The sample variance (s 2 ) is a statistic that describes the variation within the sample on the basis of the deviations of individual observations from the mean. The sample variance is equal to the sum of the squared deviations of individual observations (x) from their sample mean ( - x - ), divided by one less than the total number of observations (n): s 2 = ( - x - x) 2 n 1 t-test for comparison of sample means The two sample variances (s 2 1 and s 2 2) may be combined and used in a test of difference between the sample means, the so-called t-test. Here, the difference between the sample means is compared to the standard error of the difference (s x1 - x2 ). (s x1 - x2 ) = ( (n 1 1) s (n 2 1) s 2 2) ( ) n 1 + n 2-2 n 1 n 2 Using the standard error of the difference (s x1 - x2 ), it is possible to calculated a t- calculated (t calc ) value. The t calc value is compared to the tabled critical value (t table value), (Table 1) with 0.05 probability (95% confidence level) and n 1 + n 2-2 degrees of freedom. t calc = - x x - 2 s x1 - x2

2 The t table value is obtained from a table of critical values of the t distribution (Table 1) for the desired probability of confidence level and n 1 degrees of freedom. The statistical concept of degrees of freedom refers to the number of items that can vary independently. Once these n -1 observations are determined, the last observation is automatically set because it must be equal to the observed sample mean. The level of confidence refers to the desired probability, selected by the investigator (you), that the true mean will be included in the calculated limits. If t calc < t table, then accept the null hypothesis, (in other words, there is no statistically significant difference between the means). If t calc > t table, then reject your null hypothesis, (in other words, there is a statistically significant difference between the means). We will utilize these statistical concepts to test the null hypothesis that the difference between the calculated mean egg masses in grams is no greater than expected for two different types of eggs.

3 Steps for statistical analysis: Step 1. Calculate the sample variance (s 2 ) for Group 1 (s 2 1) and Group 2 (s 2 2) using example from Table 1., Hypothesis Testing Lab, part 1. s 2 = ( - x - x) 2 (Equation 1) n 1 Group 1 = large white eggs Mass Average egg mass (x 1 ) ( - x - 1) ( - x x 1 ) ( - x x 1 ) Group 2 = extra-large white eggs s 2 1 = ( - x x 1 ) 2 = n-1 Mass Average egg mass (x 2 ) ( - x - 2) ( - x - 2 x 2 ) ( - x - 2 x 2 ) s 2 2 = ( - x - 2 x 2 ) 2 = n-1

4 t-test for comparison of sample means The two sample variances may be combined and used in a test of difference between the sample means, the so-called t-test. Here, the difference between the sample means is compared to the standard error of the difference (s x1 - x2 ). Step 2. Calculate the standard error of the difference according to the following formula: (s x1 - x2 ) = ( (n 1 1) s (n 2 1) s 2 2) ( ) n 1 + n 2-2 n 1 n 2 Where: x 1, s 2 1, n 1 = values for Group 1 (large white eggs) x 2, s 2 2, n 2 = values for Group 2 (extra large white eggs) (s x1 - x2 ) = (11 (4.3256) + 11 (6.0789) ) ( ) = Step 3. Calculate a t value by the formula: t calc = - x x - 2 s x1 - x2 t calc = = Step 4. Compare the absolute value of calculated t value to the tabled critical value (Table 1) with 0.05 probability (95% confidence level) and n 1 + n 2-2 degrees of freedom. t table (0.95; 22) = (from Table 1) t calc = In this example, t calc > t table, > 2.074

5 Step 5. If t calc < t table, then accept the null hypothesis, (in other words, there is no statistically significant difference between the means). Conclusion: There is no statistically significant difference in the average mass in grams between large white chicken eggs and extra-large white chicken eggs. If t calc > t table, then reject your null hypothesis, (in other words, there is a statistically significant difference between the means). Conclusion: There is a statistically significant difference in the average mass in grams between large white chicken eggs and extra-large white chicken eggs.