Final Exam Version A Open Book and Notes your 4-digit code: Staple the question sheets to your answers Write your name only once on the back of this sheet. Problem 1: (10 points) A popular method to isolate single cells for further characterization is the limiting dilution method. A cell suspension is diluted to an appropriate low concentration and 0.05 ml of this suspension is dispersed into each well of a 96 well plate. What should be cell number concentration (number of cells per ml) in the dilute suspension such that not a single well out of the 96 contains two cells? At this concentration, how many wells are expected to have a single cell? Problem 2: (10 points) Potassium mass fraction (% by weight) in fertilizer samples are measured in a commercial product as shown below: 21.9 23.3 22.1 22.3 24.7 24.5 24.0 24.1 24.2 26.5 23.8 25.3 24.8 24.5 27.8 24.9 27.2 25.1 25.5 23.7 26.5 22.0 26.7 25.2 23.1 22.8 25.2 23.7 24.6 a. Construct a stem-and-leaf diagram for these data and comment on the apparent distribution. b. Find the 99 % confidence intervals for the sample variance and standard deviation. Problem 3: (5 points) The following data were obtained on total nitrogen concentration (in ppm) of water drawn from a lake being considered for use as a source of drinking water for a town.: 0.045 0.055 0.049 0.028 0.025 0.039 0.023 0.045 0.038 0.035 0.026 0.059 Find a 95 % one-sided confidence interval on the largest possible value for the mean nitrogen concentration. To be acceptable as a source of drinking water, the mean nitrogen concentration must li below 0.05 ppm. Does this lake appear to meet this criterion? Problem 4: (20 points) Another environmental monitoring company measures the total nitrogen concentration (ppm) in the same lake (as in problem above) and obtains the following data: 0.042 0.023 0.049 0.036 0.045 0.025 0.048 0.035 0.048 0.043 0.044 0.052 (a) Can the variances for these two sets of data be considered equal or unequal? (b) Are the two mean nitrogen concentrations the same or different with 95% confidence level? Problem 5: (15 points) The blood glucose concentration measured before breakfast in the normal human population is normally distributed with a mean of 100 gm/dl and known standard deviation of 10 gm/dl. At 95% confidence, what are the acceptance limits of blood glucose concentration for a normal person? What is the P value of a diabetic patient s blood glucose measurement being 135 gm/dl? If this patient s blood glucose concentration is normally distributed with a mean of 140 gm/dl and a standard deviation of 20 mg/dl, what is the probability that this patient will be misdiagnosed as normal?
Problem 6: (15 points) For the experimental data in the table below: Y 8 11 8 16 16 15 18 X 7 10 11 15 18 25 30 The sums are calculated: Σx = 116, Σy = 92, Σx 2 = 2344, Σy 2 = 1310, Σxy =1697. Find the slope and the intercept of a linear regression model. Test for the significance of regression and calculate the adjusted R 2. Problem 7: (15 points) For the experimental data in the table below, the multiple linear regression model y = β 0 + β 1 x 1 + β 2 x 2 is considered. Y 17.9 16.5 16.4 16.8 18.8 15.5 17.5 16.4 15.9 18.3 X 1 1.35 1.90 1.70 1.80 1.30 2.05 1.60 1.80 1.85 1.40 X 2 90 30 80 40 35 45 50 60 65 30 The covariance matrix is calculated and given below: 6. 071 3. 026 0. 017 C X t 1 = ( X) = 3. 026 1. 739 0. 0022 0. 017 0. 0022 0. 00026 and s 2 = 0.02. Determine the coefficients of the above model and 95% confidence intervals for the parameter β 2. Problem 8: (10 points) Construct a fractional factorial design table for six factors that requires only 8 experiments. Write down the complete defining relationship and the aliases from this design. Present the design table with a column showing the treatment conditions. Happy Holidays!
Final Exam Version B Open Book and Notes your 4-digit code: Staple the question sheets to your answers Write your name only once on the back of this sheet. Problem 1: (10 points) A popular method to isolate single cells for further characterization is the limiting dilution method. A cell suspension is diluted to an appropriate low concentration and 0.10 ml of this suspension is dispersed into each well of a 96 well plate. What should be cell number concentration (number of cells per ml) in the dilute suspension such that not a single well out of the 96 contains two cells? At this concentration, how many wells are expected to have a single cell? Problem 2: (10 points) Potassium mass fraction (% by weight) in fertilizer samples are measured in a commercial product as shown below: 24.8 24.5 27.8 24.9 27.2 25.1 25.5 23.7 23.8 25.3 22.3 24.7 24.5 24.0 24.1 24.2 26.5 21.9 23.3 22.1 26.5 22.0 26.7 25.2 23.1 22.8 25.2 23.7 24.6 25.0 a. Construct a stem-and-leaf diagram for these data and comment on the apparent distribution. b. Find the 95 % confidence intervals for the sample variance and standard deviation. Problem 3: (5 points) The following data were obtained on total nitrogen concentration (in ppm) of water drawn from a lake being considered for use as a source of drinking water for a town.: 0.042 0.023 0.049 0.036 0.045 0.025 0.048 0.035 0.048 0.043 0.044 0.052 Find a 95 % one-sided confidence interval on the largest possible value for the mean nitrogen concentration. To be acceptable as a source of drinking water, the mean nitrogen concentration must li below 0.05 ppm. Does this lake appear to meet this criterion? Problem 4: (20 points) Another environmental monitoring company measures the total nitrogen concentration (ppm) in the same lake (as in problem above) and obtains the following data: 0.045 0.055 0.049 0.028 0.025 0.039 0.023 0.045 0.038 0.035 0.026 0.059 (c) Can the variances for these two sets of data be considered equal or unequal? (d) Are the two mean nitrogen concentrations the same or different with 95% confidence level? Problem 5: (15 points) The blood glucose concentration measured before breakfast in the normal human population is normally distributed with a mean of 100 gm/dl and known standard deviation of 12 gm/dl. At 95% confidence, what are the acceptance limits of blood glucose concentration for a normal person? What is the P value of a diabetic patient s blood glucose measurement being 135 gm/dl? If this patient s blood glucose concentration is normally distributed with a mean of 140 gm/dl and a standard deviation of 20 mg/dl, what is the probability that this patient will be misdiagnosed as normal?
Problem 6: (15 points) For the experimental data in the table below: Y 10 11 8 16 16 15 18 X 9 10 11 15 18 25 30 The sums are calculated: Σx = 118, Σy = 94, Σx 2 = 2376, Σy 2 = 1346, Σxy = 1731 Find the slope and the intercept of a linear regression model. Test for the significance of regression and calculate the adjusted R 2. Problem 7: (15 points) For the experimental data in the table below, the multiple linear regression model y = β 0 + β 1 x 1 + β 2 x 2 is considered. Y 16.5 16.5 16.4 16.8 18.8 15.5 17.5 16.4 15.9 18.3 X 1 1.35 1.90 1.70 1.80 1.30 2.05 1.60 1.80 1.85 1.40 X 2 80 30 80 40 35 45 50 60 65 30 The covariance matrix is calculated and given below: 5. 899 2. 93 0. 0173 C X t 1 = ( X) = 2. 93 1. 723 0. 00086 0. 0173 0. 00086 0. 000308 and s 2 = 0.135. Determine the coefficients of the above model and 95% confidence intervals for the parameter β 2. Problem 8: (10 points) Construct a fractional factorial design table for six factors that requires only 8 experiments. Write down the complete defining relationship and the aliases from this design. Present the design table with a column showing the treatment conditions. Happy Holidays!
Final Exam Version C Open Book and Notes your 4-digit code: Staple the question sheets to your answers Write your name only once on the back of this sheet. Problem 1: (10 points) A popular method to isolate single cells for further characterization is the limiting dilution method. A cell suspension is diluted to an appropriate low concentration and 0.15 ml of this suspension is dispersed into each well of a 96 well plate. What should be cell number concentration (number of cells per ml) in the dilute suspension such that not a single well out of the 96 contains two cells? At this concentration, how many wells are expected to have a single cell? Problem 2: (10 points) Potassium mass fraction (% by weight) in fertilizer samples is measured in a commercial product as shown below: 24.5 24.0 24.1 24.2 26.5 21.9 23.3 22.1 22.3 24.7 25.1 25.5 23.7 23.8 25.3 24.8 24.5 27.8 24.9 27.2 26.7 25.2 23.1 22.8 25.2 23.7 24.6 26.5 22.0 25.0 a. Construct a stem-and-leaf diagram for these data and comment on the apparent distribution. b. Find the 90 % confidence intervals for the sample variance and standard deviation. Problem 3: (5 points) The following data were obtained on total nitrogen concentration (in ppm) of water drawn from a lake being considered for use as a source of drinking water for a town: 0.039 0.023 0.045 0.038 0.035 0.026 0.059 0.045 0.055 0.049 0.028 0.025 Find a 95 % one-sided confidence interval on the largest possible value for the mean nitrogen concentration. To be acceptable as a source of drinking water, the mean nitrogen concentration must li below 0.05 ppm. Does this lake appear to meet this criterion? Problem 4: (20 points) Another environmental monitoring company measures the total nitrogen concentration (ppm) in the same lake (as in problem above) and obtains the following data: 0.025 0.048 0.035 0.048 0.043 0.044 0.052 0.042 0.023 0.049 0.036 0.045 (e) Can the variances for these two sets of data be considered equal or unequal? (f) Are the two mean nitrogen concentrations the same or different with 95% confidence level? Problem 5: (15 points) The blood glucose concentration measured before breakfast in the normal human population is normally distributed with a mean of 100 gm/dl and known standard deviation of 14 gm/dl. At 95% confidence, what are the acceptance limits of blood glucose concentration for a normal person? What is the P value of a diabetic patient s blood glucose measurement being 135 gm/dl? If this patient s blood glucose concentration is normally distributed with a mean of 140 gm/dl and a standard deviation of 20 mg/dl, what is the probability that this patient will be misdiagnosed as normal?
Problem 6: (15 points) For the experimental data in the table below: Y 8 11 8 16 16 15 23 X 7 8 11 15 18 25 30 The sums are calculated: Σx = 114, Σy = 97, Σx 2 = 2308, Σy 2 = 1515, Σxy =1825 Find the slope and the intercept of a linear regression model. Test for the significance of regression and calculate the adjusted R 2. Problem 7 (15 points) For the experimental data in the table below, the multiple linear regression model y = β 0 + β 1 x 1 + β 2 x 2 is considered. Y 17.9 17 16.4 16.8 18.8 15.5 17.5 16.4 15.9 18.3 X 1 1.8 1.9 1.7 1.8 1.3 2.05 1.6 1.8 1.85 1.4 X 2 90 30 80 40 35 45 50 60 65 30 The matrix calculations are mostly done and the results are provided below: 6. 402 3. 59 0. 0024 C X t 1 = ( X) = 3. 59 2. 303 0. 007 0. 0024 0. 007 0. 000277 and s 2 = 0.417. Determine the coefficients of the above model and 95% confidence intervals for the parameter β 2. Problem 8 (10 points) Construct a fractional factorial design table for six factors that requires only 8 experiments. Write down the complete defining relationship and the aliases from this design. Present the design table with a column showing the treatment conditions. Happy Holidays!