Final Exam Version A

Transcription

1 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: 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.: 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: (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?

2 Problem 6: (15 points) For the experimental data in the table below: Y X 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 X X The covariance matrix is calculated and given below: C X t 1 = ( X) = and s 2 = 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!

3 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: 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.: 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: (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?

4 Problem 6: (15 points) For the experimental data in the table below: Y X 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 X X The covariance matrix is calculated and given below: C X t 1 = ( X) = and s 2 = 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!

5 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: 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: 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: (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?

6 Problem 6: (15 points) For the experimental data in the table below: Y X 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 X X The matrix calculations are mostly done and the results are provided below: C X t 1 = ( X) = and s 2 = 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!