Unit 3 Problem Set NAME: Elements of Statistics--FHSU Virtual College--Summer 2015 REMEMBER, these are assessed preparatory problems related to the content of Unit 3. The Unit 3 Exam will consist of similar types of problems, but not exactly the same. Thus, make sure you are thinking about the concepts and procedures you studied in this unit versus simply \"copying\" the process of an example problem. Also, take time to examine the complete objective list in the Unit 3 Review document. Listed out to the left of the spreadsheet are text chapter separators if you find yourself needing some direction to a related resource. All answers should be calculated, as needed, within this Excel sheet, and final concluding answers given directly below or to the right of the problem. Please make your answers are easily found--for example use a different color or type of font. No numerical answer resulting from a calculation will be accepted unless the process is performed in Excel and formulas/calculations used are evident when the cell is selected. Also, note that the templates for hypothesis testing provided in the Excel Guides for this unit are also given in the next worksheet in this document-see folder tabs at the bottom of the sheet. You may use these templates by copying from the second worksheet, pasting the copy to the right of the associated problem, then changing values as needed. Problems related to text's Chapter 7: 1. Assume you need to build a confidence interval for a population mean within some given situation. Naturally, you must determine whether you should use either the t-distribution or the z-distribution or possibly even neither based upon the information known/collected in the situation. Thus, based upon the information provided for each situation below, determine which (t-, z- or neither) distribution is appropriate. Then if you can use either a t- or z- distribution, give the associated critical value (critical t- or z- score) from that distribution to reach the given confidence level. a. 90% confidence n=150 Appropriate distribution: Associated critical value: known population data believed to be very skewed b. 95% confidence n=10 Appropriate distribution: Associated critical value: unknown population data believed to be skewed right c. 95% confidence n=40 Appropriate distribution: Associated critical value: unknown population data believed to be normally distributed d. 99% confidence n=12 Appropriate distribution: Associated critical value: unknown population data believed to be normally distributed 2. A student researcher is interested in determining the average () GPA of all FHSU students, in order to investigate grade inflation at regional universities. The data below represent the GPA's of thirty randomly selected FHSU students. 2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12 4.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95 2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14 a. How do you know that you will need to construct the confidence interval using a t-distribution approach as opposed to a zdistribution? 3.67 2.50 1.15 We want to construct the mean value confidence interval for the GPA's with a 90% confidence level. b. Determine the best point estimate (average) for the mean GPA. c. Determine the critical t-value(s) associated with the 90% confidence level. d. Determine the margin of error. e. Determine the confidence interval. f. In a sentence, interpret the contextual meaning of your result to part e above...that is relate the values to this situation regarding the mean GPA's of all FHSU students. 3. Determine the two chi-squared (2) critical values for the following confidence levels and sample sizes. a. 90% and n=60 b. 95% and n=18 4. We are also interested in estimating the population standard deviation () for all FHSU student GPA's. We will assume that GPA's are at least approximately normally distributed. Below are the GPA's. 2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12 4.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95 2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14 Out to the right, construct a 90% confidence interval estimate of sigma (), the population standard deviation. Problems related to text's Chapter 8: 5. (Multiple Choice) A hypothesis test is used to test a claim. On a right-tailed hypothesis test with a 1.39 critical value, the collected sample's test statistic is calculated to be 1.15. Which of the following is the correct decision statement for the test? A. Fail to reject the null hypothesis B. Reject the null hypothesis C. Claim the alternative hypothesis is true D. Claim the null hypothesis is false 6. (Multiple Choice) A hypothesis test is used to test a claim. A P-value of 0.23 is calculated on the hypothesis test with a significance level set at 0.05. Which of the following is the correct decision statement for the test? A. Claim the null hypothesis is true B. Claim the alternative hypothesis is false C. Reject the null hypothesis D. Fail to reject the null hypothesis 7. (Multiple Choice) Which of the following is not a requirement for using the t-distribution for a hypothesis test concerning . A. Sample size must be larger than 30 B. Sample is a simple random sample C. The population standard deviation is unknown 8. In an effort to promote healthy lifestyles, health screenings are given to employees of a large corporation. In running a promotional trial, 74 out of the 130 people who work in one office for the corporation participate in the health screening. a. Is the above information sufficient for you to be completely certain that more than 50% of all employees of the corporation will participate in the health screening? Why or why not? b. In establishing a statistical hypothesis testing of this situation, give the required null and alternative hypotheses for such a test, if it is desired that more than 50% of the employees participate in the health screening. H0: H1: c. Based on your answer in part b, should you use a right-tailed, a left-tailed, or a two-tailed test? Briefly explain how one determines which of the three possibilities is to be used. d. Describe the possible Type I error for this situation--make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings. e. Describe the possible Type II error for this situation--make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings. f. Determine the appropriate critical value(s) for this situation given a 0.025 significance level. g. Determine/calculate the value of the sample's test statistic. h. Detemine the P-value. i. Based upon your work above, is there statistically sufficient evidence in this sample to support that more than 50% of employees will participate in the health screening? Briefly explain your reasoning. 9. The mean score on a certain achievement test at the turn of the century was 73. However, national standards have been implmented which may lead to a change in the mean score. A random sample of 32 scores on this exam taken this year yeilded the following data set. At a 10% significance level, test the claim that the mean of all current test scores is not the same as in 2000. 85 77 74 88 89 66 0 70 58 76 86 89 73 82 72 0 82 82 80 76 87 76 77 67 72 49 73 75 82 18 81 30 a. Give the null and alternative hypotheses for this test in symbolic form. H0: H1: b. Determine the value of the test statistic. c. Determine the appropriate critical value(s). d Detemine the P-value. e. Is there sufficient evidence to support the claim that the mean achivement score is now different than 73? Explain your reasoning. Problem related to text's Chapter 9: 10. Listed below are pretest and posttest scores from a study. Using a 5% significance level, is there statistically sufficient evidence to support the claim that the posttest scores were higher than the pretest scores? Perform an appropriate hypothesis test showing necessary statistical evidence to support your final given conclusion. PreTest 24 11 14 25 17 28 22 PostTest 28 16 18 27 15 31 21 Problems related to text's Chapter 10: 11. Multiple Choice: For each of the following data sets, choose the most appropriate response from the choices below the table. Data Set #1 Data Set #2 x y x y 0 19 10 100 1 15 14 33 2 13 18 124 3 12 24 160 4 7 27 65 5 0 32 117 6 -3 36 27 7 -4 40 150 8 -7 45 44 A. A strong positive linear relation exists A. A strong positive linear relation exists B. A strong negative linear relation exists B. A strong negative linear relation exists C. A curvilinear relation exists C. A curvilinear relation exists D. No linear relation exists D. No linear relation exists 12. Create a paired data set with 5 data points indicating strong (but not perfect) positive linear correlation. Determine the correlation coefficient value for your data x y 13. To answer the following, use the given data that contains information on the age of eight randomly female staff members at FHSU and their corresponding pulse rate. Age (years) Pulse Rate (BPM) 42 34 49 27 42 18 41 21 98 80 98 63 84 49 80 55 a. Construct a scatterplot for this data set in the region to the right (age as the independent variable, and pulse rate as the dependent.) b. Based on the scatterplot, does it look like a linear regression model is appropriate for this data? Why or why not? c. Add the line-of-best fit (trend line/linear regression line) to your scatterplot. Give the equation of the trend line below. Then give the slope value of the line and explain its meaning to this context. d. Determine the value of the correlation coefficient. Explain what the value tells you about the data pairs? e. Does the value of the correlation coefficient tell you there is or is not statistically significant evidence that correlation exists between the age and pulse rates of female staff members? Explain your position. (HINT: application of table A-6 is needed!) f. Based on the above, what is the best predicted pulse rate of a 30 year old female staff member? 3.67 2.50 1.15 Templates for Hypothesis Testing As stated on the practice exam document, below you will find templates you may use in completing this exam. You will want to highlight the one you need, copy it to where you are working, and then input the proper values for the problem you are working on for quantities related to the labels in red. Single Sample Proportion Test: Two-tailed Proportion significance level (alpha) p= x= n= One-tailed Proportion significance level (alpha) p= x= n= 0.01 0.82 56 73 q= 0.18 p-hat = 0.7671232877 critical values are: #NAME? and test statistic = -1.175933319 P-value = #NAME? 0.01 0.82 56 73 q= 0.18 p-hat = 0.7671232877 Left Tailed Rt. Tailed critical value is: #NAME? or #NAME? test statistic = -1.175933319 P-value = #NAME? #NAME? Single Sample Mean Test (or difference in matched paired two samples) Two-tailed Mean (sigma known) significance level (alpha) x-bar = mu, = sigma, = n= critical values are: test statistic = P-value = 0.05 24.85 24 2 25 #NAME? and 2.125 #NAME? #NAME? Two-tailed Mean (sigma unknown) significance level (alpha) 0.05 x-bar = 110 mu, = 118 s= 12 n= 20 critical values are: test statistic = P-value = #NAME? and -2.98142397 #NAME? #NAME? One-tailed Mean (sigma known) significance level (alpha) 0.05 x-bar = 24.85 mu, = 24 sigma, = 2 n= 25 Left Tailed Rt. Tailed critical value is: #NAME? or #NAME? test statistic = 2.125 P-value = #NAME? One-tailed Mean (sigma unknown) significance level (alpha) 0.05 x-bar = 110 mu, = 118 s= 12 n= 20 Left Tailed Rt. Tailed critical value is: #NAME? or #NAME? test statistic = -2.98142397 P-value = #NAME? Single Sample Variance/Standard Deviation Test Two-tailed Standard Deviation significance level (alpha) 0.05 n= 25 s= 0.029 = 0.023 s2 = 0.000841 2 = 0.000529 critical values are: #NAME? and test statistic = 38.155009452 P-value = #NAME? #NAME? One-tailed Standard Deviation significance level (alpha) 0.05 n= 25 s= 0.029 = 0.023 s2 = 0.000841 2 = 0.000529 Left Tailed Rt. Tailed critical values are: #NAME? or #NAME? test statistic = 38.155009452 P-value = #NAME? Inferences About 2 Proportions Two-tailed Proportion (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x= 56 27 n= 843 703 alpha, = One-tailed Proportion (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x= 50 16 n= 290 123 0.05 alpha, = p-bar = 0.053686934 q-bar = 0.946313066 critical values are: #NAME? and #NAME? test statistic = 2.4341272112 P-value = #NAME? 0.05 p-bar = 0.1598062954 q-bar = 0.8401937046 critical value is: #NAME? or #NAME? test statistic = 1.0736524347 P-value = #NAME? Inferences about 2 Means: Independent Samples Two-tailed Mean (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x-bar = 4.2 1.71 n= 22 22 s= 2.2 0.72 alpha, = 0.05 critical values are: #NAME? and #NAME? test statistic = 5.0453711835 P-value = #NAME? One-tailed Mean (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x-bar = 0.94 1.65 n= 21 8 s= 0.31 0.16 alpha, = 0.05 Right Tail Left Tail critical value is: #NAME? or #NAME? test statistic = -8.051464895 P-value = #NAME? Unit 3 Problem Set NAME: Elements of Statistics--FHSU Virtual College--Summer 2015 REMEMBER, these are assessed preparatory problems related to the content of Unit 3. The Unit 3 Exam will consist of similar types of problems, but not exactly the same. Thus, make sure you are thinking about the concepts and procedures you studied in this unit versus simply \"copying\" the process of an example problem. Also, take time to examine the complete objective list in the Unit 3 Review document. Listed out to the left of the spreadsheet are text chapter separators if you find yourself needing some direction to a related resource. All answers should be calculated, as needed, within this Excel sheet, and final concluding answers given directly below or to the right of the problem. Please make your answers are easily found--for example use a different color or type of font. No numerical answer resulting from a calculation will be accepted unless the process is performed in Excel and formulas/calculations used are evident when the cell is selected. Also, note that the templates for hypothesis testing provided in the Excel Guides for this unit are also given in the next worksheet in this document--see folder tabs at the bottom of the sheet. You may use these templates by copying from the second worksheet, pasting the copy to the right of the associated problem, then changing values as needed. Problems related to text's Chapter 7: 1. Assume you need to build a confidence interval for a population mean within some given situation. Naturally, you must determine whether you should use either the tdistribution or the z-distribution or possibly even neither based upon the information known/collected in the situation. Thus, based upon the information provided for each situation below, determine which (t-, z- or neither) distribution is appropriate. Then if you can use either a t- or z- distribution, give the associated critical value (critical t- or zscore) from that distribution to reach the given confidence level. a. 90% confidence n=150 known Appropriate distribution: Z distribution Associated critical value: 1.645 population data believed to be very skewed b. 95% confidence n=10 unknown Appropriate distribution: Neither Associated critical value: population data believed to be skewed right c. 95% confidence n=40 unknown Appropriate distribution: z distribution Associated critical value: 1.96 population data believed to be normally distributed d. 99% confidence n=12 Appropriate distribution: Associated critical value: population data believed to be normally distributed unknown t distribution 3.106 2. A student researcher is interested in determining the average () GPA of all FHSU students, in order to investigate grade inflation at regional universities. The data below represent the GPA's of thirty randomly selected FHSU students. 2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12 4.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95 2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14 a. How do you know that you will need to construct the confidence interval using a t-distribution approach as opposed to a z-distribution? 3.67 2.50 1.15 As we do not know population standard deviation and sample size is not greater than 30 so we will use t distribution instead of z distribution. We want to construct the mean value confidence interval for the GPA's with a 90% confidence level. b. Determine the best point estimate (average) for the mean GPA. 2.688 c. Determine the critical t-value(s) associated with the 90% confidence level. #NAME? d. Determine the margin of error. #NAME? e. Determine the confidence interval. Confidence interval is from 2.688 - 0.277 to 2.688 + 0.277 ie 2.411 to 2.965 f. In a sentence, interpret the contextual meaning of your result to part e above...that is relate the values to this situation regarding the mean GPA's of all FHSU students. We are 90% confident that mean GPA's of all FHSU students lie in the interval 2.411 to 2.965 3. Determine the two chi-squared (2) critical values for the following confidence levels and sample sizes. a. 90% and n=60 b. 95% and n=18 4. We are also interested in estimating the population standard deviation () for all FHSU student GPA's. We will assume that GPA's are at least approximately normally distributed. Below are the GPA's. 2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12 4.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95 2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14 Out to the right, construct a 90% confidence interval estimate of sigma (), the population standard deviation. 3.67 2.50 1.15 Problems related to text's Chapter 8: 5. (Multiple Choice) A hypothesis test is used to test a claim. On a right-tailed hypothesis test with a 1.39 critical value, the collected sample's test statistic is calculated to be 1.15. Which of the following is the correct decision statement for the test? A. Fail to reject the null hypothesis B. Reject the null hypothesis C. Claim the alternative hypothesis is true D. Claim the null hypothesis is false 6. (Multiple Choice) A hypothesis test is used to test a claim. A P-value of 0.23 is calculated on the hypothesis test with a significance level set at 0.05. Which of the following is the correct decision statement for the test? A. Claim the null hypothesis is true B. Claim the alternative hypothesis is false C. Reject the null hypothesis D. Fail to reject the null hypothesis 7. (Multiple Choice) Which of the following is not a requirement for using the t-distribution for a hypothesis test concerning . A. Sample size must be larger than 30 B. Sample is a simple random sample C. The population standard deviation is unknown 8. In an effort to promote healthy lifestyles, health screenings are given to employees of a large corporation. In running a promotional trial, 74 out of the 130 people who work in one office for the corporation participate in the health screening. a. Is the above information sufficient for you to be completely certain that more than 50% of all employees of the corporation will participate in the health screening? Why or why not? b. In establishing a statistical hypothesis testing of this situation, give the required null and alternative hypotheses for such a test, if it is desired that more than 50% of the employees participate in the health screening. H0: H1: c. Based on your answer in part b, should you use a right-tailed, a left-tailed, or a two-tailed test? Briefly explain how one determines which of the three possibilities is to be used. d. Describe the possible Type I error for this situation--make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings. e. Describe the possible Type II error for this situation--make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings. f. Determine the appropriate critical value(s) for this situation given a 0.025 significance level. g. Determine/calculate the value of the sample's test statistic. h. Detemine the P-value. i. Based upon your work above, is there statistically sufficient evidence in this sample to support that more than 50% of employees will participate in the health screening? Briefly explain your reasoning. 9. The mean score on a certain achievement test at the turn of the century was 73. However, national standards have been implmented which may lead to a change in the mean score. A random sample of 32 scores on this exam taken this year yeilded the following data set. At a 10% significance level, test the claim that the mean of all current test scores is not the same as in 2000. 85 77 74 88 89 66 0 70 58 76 86 89 73 82 72 0 82 82 80 76 87 76 77 67 72 49 73 75 82 18 81 30 a. Give the null and alternative hypotheses for this test in symbolic form. H0: H1: b. Determine the value of the test statistic. c. Determine the appropriate critical value(s). d Detemine the P-value. e. Is there sufficient evidence to support the claim that the mean achivement score is now different than 73? Explain your reasoning. Problem related to text's Chapter 9: 10. Listed below are pretest and posttest scores from a study. Using a 5% significance level, is there statistically sufficient evidence to support the claim that the posttest scores were higher than the pretest scores? Perform an appropriate hypothesis test showing necessary statistical evidence to support your final given conclusion. PreTest 24 11 14 25 17 28 22 PostTest 28 16 18 27 15 31 21 Problems related to text's Chapter 10: 11. Multiple Choice: For each of the following data sets, choose the most appropriate response from the choices below the table. Data Set #1 Data Set #2 x y x y 0 19 10 100 1 15 14 33 2 13 18 124 3 12 24 160 4 7 27 65 5 0 32 117 6 -3 36 27 7 -4 40 150 8 -7 45 44 A. A strong positive linear relation exists A. A strong positive linear relation exists B. A strong negative linear relation exists B. A strong negative linear relation exists C. A curvilinear relation exists C. A curvilinear relation exists D. No linear relation exists D. No linear relation exists 12. Create a paired data set with 5 data points indicating strong (but not perfect) positive linear correlation. Determine the correlation coefficient value for your data x y 1 2 2 3.98 Column 1 Column 2 3 6 Column 1 1 4 7.99 Column 2 0.9999920697 1 5 9.96 Correlation coefficient 13. To answer the following, use the given data that contains information on the age of eight randomly female staff members at FHSU and their corresponding pulse rate. Age (years) Pulse Rate (BPM) 42 34 49 27 42 18 41 21 98 80 98 63 84 49 80 55 100 90 80 Pulse RAT E (BPM) 70 60 50 a. 40 Construct a scatterplot for this data set in the region to the right (age as the independent variable, and pulse rate as the dependent.) 20 b. Based on the scatterplot, does it look like a linear regression model is appropriate for this data? Why or why not? Data appears to be linearly related so linear regression model is appropriate for this data Column 2 1 This coefficient tells about the strength of linear relationship between Age and Pulse Rate e. Does the value of the correlation coefficient tell you there is or is not statistically significant evidence that correlation exists between the age and pulse rates of female staff members? Explain your position. (HINT: application of table A-6 is needed!) Sample size(n) = 8.Degree of freedom = n-2 = 6 .Critical T value = 1.9432 value = r * sqrt[(n-2)/(1-r2)] = 0.9613 * sqrt(6/1-0.9613 2) = 8.547 As t value > Critical T value, so there is statistically significant evidence that correlation exists between the age and pulse rate of female staff members f. Based on the above, what is the best predicted pulse rate of a 30 year old female staff member? Predicted pulse rate = 22.331 + 1.593 * 30 = 70.121 BPM 30 35 Age (year s) c. Add the line-of-best fit (trend line/linear regression line) to your scatterplot. Give the equation of the trend line below. Then give the slope value of the line and explain its meaning to this context. Equation of trend line is : Pulse Rate (BPM) = 22.331 + 1.593 * Age (years) Slope = 1.593 and it means that with increase in age by 1 year, Pulse Rate increases by 1.593 BPM. d. Determine the value of the correlation coefficient. Explain what the value tells you about the data pairs? Column 1 Column 1 1 Correlation Coefficient r Column 2 0.9612735409 25 t #NAME? 40 45 50 Templates for Hypothesis Testing As stated on the practice exam document, below you will find templates you may use in completing this exam. You will want to highlight the one you need, copy it to where you are working, and then input the proper values for the problem you are working on for quantities related to the labels in red. Single Sample Proportion Test: Two-tailed Proportion significance level (alpha) p= x= n= One-tailed Proportion significance level (alpha) p= x= n= 0.01 0.82 56 73 q= 0.18 p-hat = 0.7671232877 critical values are: #NAME? and test statistic = -1.175933319 P-value = #NAME? 0.01 0.82 56 73 q= 0.18 p-hat = 0.7671232877 Left Tailed Rt. Tailed critical value is: #NAME? or #NAME? test statistic = -1.175933319 P-value = #NAME? #NAME? Single Sample Mean Test (or difference in matched paired two samples) Two-tailed Mean (sigma known) significance level (alpha) x-bar = mu, = sigma, = n= critical values are: test statistic = P-value = 0.05 24.85 24 2 25 #NAME? and 2.125 #NAME? #NAME? Two-tailed Mean (sigma unknown) significance level (alpha) 0.05 x-bar = 110 mu, = 118 s= 12 n= 20 critical values are: test statistic = P-value = #NAME? and -2.98142397 #NAME? #NAME? One-tailed Mean (sigma known) significance level (alpha) 0.05 x-bar = 24.85 mu, = 24 sigma, = 2 n= 25 Left Tailed Rt. Tailed critical value is: #NAME? or #NAME? test statistic = 2.125 P-value = #NAME? One-tailed Mean (sigma unknown) significance level (alpha) 0.05 x-bar = 110 mu, = 118 s= 12 n= 20 Left Tailed Rt. Tailed critical value is: #NAME? or #NAME? test statistic = -2.98142397 P-value = #NAME? Single Sample Variance/Standard Deviation Test Two-tailed Standard Deviation significance level (alpha) 0.05 n= 25 s= 0.029 = 0.023 s2 = 0.000841 2 = 0.000529 critical values are: #NAME? and test statistic = 38.155009452 P-value = #NAME? #NAME? One-tailed Standard Deviation significance level (alpha) 0.05 n= 25 s= 0.029 = 0.023 s2 = 0.000841 2 = 0.000529 Left Tailed Rt. Tailed critical values are: #NAME? or #NAME? test statistic = 38.155009452 P-value = #NAME? Inferences About 2 Proportions Two-tailed Proportion (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x= 56 27 n= 843 703 alpha, = One-tailed Proportion (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x= 50 16 n= 290 123 0.05 alpha, = p-bar = 0.053686934 q-bar = 0.946313066 critical values are: #NAME? and #NAME? test statistic = 2.4341272112 P-value = #NAME? 0.05 p-bar = 0.1598062954 q-bar = 0.8401937046 critical value is: #NAME? or #NAME? test statistic = 1.0736524347 P-value = #NAME? Inferences about 2 Means: Independent Samples Two-tailed Mean (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x-bar = 4.2 1.71 n= 22 22 s= 2.2 0.72 alpha, = 0.05 critical values are: #NAME? and #NAME? test statistic = 5.0453711835 P-value = #NAME? One-tailed Mean (w/ two Ind. Samples) Given info: From Sample #1 From Sample #2 x-bar = 0.94 1.65 n= 21 8 s= 0.31 0.16 alpha, = 0.05 Right Tail Left Tail critical value is: #NAME? or #NAME? test statistic = -8.051464895 P-value = #NAME