Testing of Hypothesis

Questions: 1. Select an appropriate test of hypothesis to determine if the mean water temperature is different between the two farms. You should include the following: (a) State the null and alternative hypotheses and explain how you developed these two hypotheses.(b) Use a suitable computer software and present the output from your software for an appropriate hypothesis test with = 0.05. List the test statistic you will use for this test. Explain your decision criteria. (c) Together with your earlier findings on revenue, execute your hypothesis test and conclude if water temperature seems to be a probable cause. (d) Discuss the assumptions you have made in performing the hypothesis test. Are these assumptions reasonable for this case? Please provide explanation/s to substantiate your view. 2. Highlight three (3) possible business or statistical concerns regarding the approach used here in determining whether the mean revenue is significantly different. If your group has a free hand on the pro ject, starting from measurements, sampling, to analysis, what recommendations would you give for the problem in question? Keep your answer within 400 to 500 words (indicate your word count). You must demonstrate well developed written proficiency. 3. Provide an executive summary of your analysis and findings on this matter to Mr. Suhaimi. Answers: 1. a) The null and alternative hypothesis is given as below: Null hypothesis: H0: The mean water temperature is same for two different farms. Alternative hypothesis: Ha: The mean water temperature is not same for two different farms. We can also write these hypotheses as below: Null hypothesis: H0: There is no any significant difference in the two different farms. Alternative hypothesis: Ha: There is significant difference in the two different farms. Statistically, we can write these hypotheses as below: H0: 1 = 2 versus Ha: 1 2 Here, we have to test the research question under study whether the mean water temperature is same for given types of farms or not. For this purpose, we developed these types of hypothesis. b) We used excel software for analysis for this test. Here we have to use the two sample t test for the two population means. The excel output for this test is given below: Calculations Area Pop. 1 Sample Variance 2.1102 Pop. 2 Sample Variance 1.3005 Pop. 1 Sample Var./Sample Size 0.0211 Pop. 2 Sample Var./Sample Size 0.0130 For one-tailed tests: TDIST value 0.0000 1-TDIST value 1.0000 Two sample t test (assumes unequal population variances) Data Hypothesized Difference 0 Level of Significance 0.05 Population 1 Sample Sample Size 100 Sample Mean 22.97 Sample Standard Deviation 1.4527 Population 2 Sample Sample Size 100 Sample Mean 24.45 Sample Standard Deviation 1.1404 Intermediate Calculations Numerator of Degrees of Freedom 0.0012 Denominator of Degrees of Freedom 0.0000 Total Degrees of Freedom 187.4364 Degrees of Freedom 187 Standard Error 0.1847 Difference in Sample Means -1.4800 Separate-Variance t Test Statistic -8.0138 Two-Tail Test Lower Critical Value -1.9727 Upper Critical Value 1.9727 p-Value 0.0000 Reject the null hypothesis The sample mean for Bintan farm water temperature is observed as 22.97 units with the standard deviation 1.4527 units. The sample mean for Pinang farm water temperature is observed as 24.45 units with the standard deviation 1.1404. Group Statistics Farm N Mean Std. Deviation Std. Error Mean Water temperature Bintan 100 22.9700 1.45265 .14527 Pinang 100 24.4500 1.14040 .11404 Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Water temperature Equal variances assumed 8.838 .003 -8.014 198 .000 -1.48000 .18468 -1.84419 -1.11581 Equal variances not assumed -8.014 187.436 .000 -1.48000 .18468 -1.84432 -1.11568 c) Here, we have to take decision regarding the null hypothesis whether we have to reject the null hypothesis or not. We get the following p-value for this test. Two-Tail Test Lower Critical Value -1.9727 Upper Critical Value 1.9727 p-Value 0.0000 Reject the null hypothesis The decisions rule for rejecting or do not rejecting the null hypothesis is given as below: If the p-value is less than the given level of significance or alpha value, then we do not reject the null hypothesis and if the p-value is greater than the given level of significance or alpha value, then we reject the null hypothesis. Here, we get the p-value as 0.000. This means, the p-value is less than the given level of significance or alpha value 0.05. So, we reject the null hypothesis that there is no any significant difference between the two farms or average or mean water temperature is same for two farms. d) We made the assumption that the populations for the both farms have normal distribution. We assume this assumption because for the large number of population for any variable, data have tendency to follow an approximate normal distribution. 2. Here we have to see different three possible statistical concerns regarding the approach used here in determining whether the mean revenue is significantly different for two farms. For checking this hypothesis, we have to use the two sample t test. The null and alternative hypothesis for this test is given as below: Null hypothesis: H0: The mean revenue is same for two different farms. Alternative hypothesis: Ha: The mean revenue is not same for two different farms. We can also write these hypotheses as below: Null hypothesis: H0: There is no any significant difference for mean revenue for given two different farms. Alternative hypothesis: Ha: There is significant difference for mean revenue for given two different farms. Statistically, we can write these hypotheses as below: H0: 1 = 2 versus Ha: 1 2 The statistical output for this test is given below: Calculations Area Pop. 1 Sample Variance 569656.4426 Pop. 2 Sample Variance 227399.9699 Pop. 1 Sample Var./Sample Size 5696.5644 Pop. 2 Sample Var./Sample Size 2273.9997 For one-tailed tests: TDIST value 0.0000 1-TDIST value 1.0000 Two sample t test (assumes unequal population variances) Data Hypothesized Difference 0 Level of Significance 0.05 Population 1 Sample Sample Size 100 Sample Mean 3628.998 Sample Standard Deviation 754.7559 Population 2 Sample Sample Size 100 Sample Mean 2936.502 Sample Standard Deviation 476.8647 Intermediate Calculations Numerator of Degrees of Freedom 63529892.4735 Denominator of Degrees of Freedom 380019.4030 Total Degrees of Freedom 167.1754 Degrees of Freedom 167 Standard Error 89.2780 Difference in Sample Means 692.4960 Separate-Variance t Test Statistic 7.7566 Two-Tail Test Lower Critical Value -1.9743 Upper Critical Value 1.9743 p-Value 0.0000 Reject the null hypothesis The sample mean for Bintan farm revenue is observed as $3628.998 units with the standard deviation $754.7559. The sample mean for Pinang farm revenue is observed as $2936.502 with the standard deviation $476.8647. For this t test we get the p-value as 0.000 which is less than the given level of significance 0.05, so we reject the null hypothesis that there is no any significant difference for mean revenue for given two different farms. Now, we have to test another hypothesis which is given below: Null hypothesis: H0: The mean other revenue is same for two different farms. Alternative hypothesis: Ha: The mean other revenue is not same for two different farms. We can also write these hypotheses as below: Null hypothesis: H0: There is no any significant difference for mean other revenue for given two different farms. Alternative hypothesis: Ha: There is significant difference for mean other revenue for given two different farms. Statistically, we can write these hypotheses as below: H0: 1 = 2 versus Ha: 1 2 The statistical output for this t test is given as below: Calculations Area Pop. 1 Sample Variance 104401.9724 Pop. 2 Sample Variance 79930.6597 Pop. 1 Sample Var./Sample Size 1044.0197 Pop. 2 Sample Var./Sample Size 799.3066 For one-tailed tests: TDIST value 0.0000 1-TDIST value 1.0000 Two sample t test (assumes unequal population variances) Data Hypothesized Difference 0 Level of Significance 0.05 Population 1 Sample Sample Size 100 Sample Mean 1071.475 Sample Standard Deviation 323.1129 Population 2 Sample Sample Size 100 Sample Mean 880.381 Sample Standard Deviation 282.7201 Intermediate Calculations Numerator of Degrees of Freedom 3397851.9270 Denominator of Degrees of Freedom 17463.3154 Total Degrees of Freedom 194.5708 Degrees of Freedom 194 Standard Error 42.9340 Difference in Sample Means 191.0940 Separate-Variance t Test Statistic 4.4509 Two-Tail Test Lower Critical Value -1.9723 Upper Critical Value 1.9723 p-Value 0.0000 Reject the null hypothesis The sample mean for Bintan farm other revenue is observed as $1071.475 with the standard deviation $323.1129. The sample mean for Pinang farm other revenue is observed as $880.381 with the standard deviation $282.7201. Here, we get the p-value as 0.000 which is less than the given level of significance or alpha value 0.05, so we reject the null hypothesis that There is no any significant difference for mean other revenue for given two different farms. 3. Summary for Statistical analysis:For testing the different claims regarding the data about the two farms namely Bintan and Pinang, we used some descriptive statistics and testing of hypothesis for testing these claims. We observed the following main findings regarding the data about the given two farms:1) The sample mean for Bintan farm water temperature is observed as 22.97 units with the standard deviation 1.4527 units. 2) The sample mean for Pinang farm water temperature is observed as 24.45 units with the standard deviation 1.1404. 3) For checking the significant difference for the average water temperature for two farms viz. Bintan and Pinang, We reject the null hypothesis that there is no any significant difference between the two farms or average or mean water temperature is same for two farms. 4) The sample mean for Bintan farm revenue is observed as $3628.998 with the standard deviation $754.7559. 5) The sample mean for Pinang farm revenue is observed as $2936.502 with th e standard deviation $476.8647. 6) For checking the significant difference for the mean revenue for two farms viz. Bintan and Pinang, We reject the null hypothesis that there is no any significant difference for mean revenue for given two different farms. 7) The sample mean for Bintan farm other revenue is observed as $1071.475 with the standard deviation $323.1129. 8) The sample mean for Pinang farm other revenue is observed as $880.381 with the standard deviation $282.7201.9) For checking the significant difference for mean other revenue for two farms viz. Bintan and Pinang, We reject the null hypothesis that There is no any significant difference for mean other revenue for given two different farms. 10) Based on the sum and mean for both Bintan and Pinang farms, Bintan has a higher earnings compared to Pinang. 11) Pinang has a lower variance and standard deviation, which means that Pinang farm, has more consistent earnings compared to Bintan.