Can someone pls answer 18-20 ?

In this lab we will perform testing of hypotheses related to the population mean. This material is in Section 13.2. Download the worksheet for this lab, fill.mtw, from LON-CAPA to your computer. Three national bottling companies distribute cans of soft drinks labeled as containing 16 oz. We take the view that, to save money, they may put slightly less in each can, on average, so we'll test H0 : = 16 vs. Ha : < 16, where denotes the mean fill, using = 0.05. Let us assume that in fact company A fills at 16 oz, but that B and C consistently put in a bit less: suppose their three respective mean fills are A = 16 oz, B = 15.95 oz, and C = 15.94 oz, and all three fill distributions have population standard deviation = 0.1. We will use the information to see how the hypotheses tests perform, but not in conducting the tests. We randomly sampled n = 25 cans from each company in each of 50 states, and stored the resulting data in the file fill.mtw. Use File>Open Worksheet to load the data from fill.mtw. Data for company A are in columns c1-c50, B in c51- c100, and C in c101-c150. Columns c1, c51, and c101 give data from state 1 (for companies A, B, and C respectively). Columns c2, c52 and c102 give the data from state 2 (for companies A, B, and C, respectively), and so on. Since company A is filling correctly, we make type I error if we reject H0 for it. The significance level, = .05, is the theoretical probability of making a type I error when using the test on samples from company A. On the other hand, we make type II error if we retain H0 for companies B and C, since they are under-filling. The symbol is used to denote the probability of making type II error. Since depends on the mean of the distribution from which we sample, will be different for companies B and C. We will compare those probabilities with the observed proportion of times such errors were made across the 50 states. A. In this part consider company A. We test fill levels across all 50 states, and then check how many states incorrectly reject the null hypothesis. 1. Which hypothesis is true for company A, H0 or Ha? [Hint: This is easy.] _________________________ 2. If we retain H0 for company A, then we make (circle one) a correct decision / type II error. 3. If we reject H0 for company A, then we make (circle one) a correct decision / type I error. Use Stat>Basic Stat>1-Sample T, samples are in columns c1-c50. Check the field "perform hypothesis test", and enter value 16 into the "hypothesized mean" field. We would like to test H0 : = 16 vs. Ha : < 16. Click on Options and select the alternative "mean is < hypothesized mean". The results will be displayed in the session window. To avoid printing delays, you are given a printed copy of all Minitab output at the end of the lab paper. For the rest of this step, use that part of the Minitab output for company A. Check Minitab's work for the state # 10, c10, for company A. Use the values of sample size (n), sample mean ( x ) and sample standard deviation (s) listed on Minitab's output under N, Mean, StDev to calculate the standard error of the sample mean and test statistic using formulas provided below. The null (hypothesized) value of the sample mean in the hypothesis test is 0 16. 4. Sample mean x _________________________ 5. Standard error = s/ n ____________________________________________ 6. Test statistic s n x t / 0 ______________________________________________________ Use Table A.3 in the textbook to find the p-value. In the top row of Table A.3 (p.693, text), find the value closest to the absolute value of the test statistic. In the column that has that value of the test statistic t, go to the row corresponding to 24 degrees of freedom (df). The table gives the amount in the right tail corresponding to the absolute value of the test statistic. For the left-tailed alternative, the p-value is the amount in the tail to the left of the value of the test statistic. 7. From Table A.3, the p-value = ________________________ On Minitab's printout the test statistic is reported under T, and p-value is reported under P. Are the values you found in questions 5 and 6 the same as the ones reported by Minitab? _____ Tests are based on random sample data, so even though company A is filling at 16oz. on average, in some states our samples might have found a high number of cans not quite full, leading us to incorrectly reject H0. When we specify = 0.05, we are taking that value of as our probability of type I error. 8. How many of the 50 states are expected to incorrectly reject H0 for company A? [Hint: If you play the lottery 50 times and each time the chance of winning is 0.05, how many times do you expect to win?]________________ Note: since the question is about what is expected (on average), the answer does not have to be an integer. Type I error: List here and mark on the data sheet those columns (states) in the first 50, where data gave p-values less than = 0.05, resulting in rejection of H0__________________________ 9. For company A, in how many states do we incorrectly reject H0.? _____________ B. In this part we only consider company B. We test fill levels across all 50 states, and then we check how many states incorrectly retain the null hypothesis. 10. Which hypothesis is true for company B, H0 or Ha? ____________________________ 11. If we retain H0 for company B, then we make (circle one) a correct decision / type II error. 12. If we reject H0 for company B, then we make (circle one) a correct decision / type I error. Use Stat>Basic Stat>1-Sample T, samples are in columns c51-c100. Check the field "perform hypothesis test", and enter value 16 into the "hypothesized mean" field. We would like to test H0 : = 16 vs. Ha : < 16. Click on Options and select the alternative "less than". The results will be displayed in the session window. To avoid printing delays, you are given a printed copy of all Minitab output at the end of the lab paper. For the rest of this step, use that part of the Minitab output for company B. Tests are based on random sample data, so even though company B is filling at 15.95oz on average, in some states our samples might have found a high number of full cans, leading us to incorrectly retain H0. The chance of this happening is the probability of type II error denoted by . Technical calculation of is outside of the scope of this introductory course, so you will not be asked to calculate , but you will use the calculated value to answer other questions. The logic of the calculation of is provided below. Using the value of the population mean for company B, B = 15.95, the null hypothesis value 16, we try to detect the mean difference of 16-15.95=0.05. We reject the null hypothesis (detect this difference) if the p-value is less than 0.05, and the probability of this even happening can be computed. For B = 15.95, this probability is about .78, and it is called statistical power. Probability of type II error is not rejecting the null hypothesis, so the probability of type II error=1-power. For company B the probability of type II error =1-0.78=0.22. 13. How many of the 50 states are expected to incorrectly retain H0 for company B? [Hint: If you play the lottery 50 times and each time the chance of winning is 0.22, how many times do you expect to win?]______________ Note: since the question is about what is expected (on average), the answer does not have to be an integer. List here and mark on the data sheet the states whose data for company B produced p-values more than =0.05, so that they incorrectly retained H0. __________________________________________________________________________________ Note: in this list, use state names c51 through c100. 14. For company B, in how many states do we incorrectly retain the null hypothesis? ___________________ C. In this part we only consider company C. We test fill levels across all 50 states, and then we check how many states incorrectly retain the null hypothesis. 15. Which hypothesis is true for company C, H0 or Ha?_____________________ 16. If we retain H0 for company C, then we make (circle one) a correct decision / type II error. 17. If we reject H0 for company C, then we make (circle one) a correct decision / type I error. Use Stat>Basic Stat>1-Sample T, samples are in columns c101-c150. Check the field "perform hypothesis test", and enter value 16 into the "hypothesized mean" field. We would like to test H0 : = 16 vs. Ha : < 16. Click on Options and select the alternative "less than". The results will be displayed in the session window. To avoid printing delays, you are given a printed copy of all Minitab output at the end of the lab paper. For the rest of this step, use that part of the Minitab output for company C. Tests are based on random sample data, so even though company C is filling at 15.94oz on average, in some states our samples might have found a high number of full cans, leading us to incorrectly retain H0. The chance of this happening, , is the probability of type II error. Using the value of the population mean for company C, C = 15.94, the null hypothesis value 16, we try to detect the mean difference of 16-15.94=.06. This difference is detected, or the null hypothesis is rejected with probability .90, so for company C the probability of type II error is =1-0.90=0.10. 18. How many of the 50 states are expected to incorrectly retain H0 for company C? [Hint: If you play the lottery 50 times and each time the chance of winning is 0.10, how many times do you expect to win?]______________ Note: since the question is about what is expected (on average), the answer does not have to be an integer. List here and mark on the data sheet the 50 states whose data for company C produced p-values more than = 0.05, so that they incorrectly retained H0? ___ _________________________________________________________________ Note: in this list, use state names c101 through c150. 19. For company C, in how many states do we incorrectly retain the null hypothesis? ________________________ D: Transcribe your findings to the table below. Company A Company B Company C = .05 =.22 = .10 Observed number of errors Expected number of errors: 20. Our test of left-sided hypothesis made fewer errors when testing for company B C. Circle one company and explain why