Answers Problem Set 2 Chem 314A Williamsen Spring 2000

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3) You are performing a quantitative analysis to determine the concentration of Ca in a sample of seawater. After determining that the other constituents of the seawater will not interfere with the analysis, you make a calibration curve from intensity data measured while analyzing samples of known Ca concentration. Then you analyze your sea water sample. After analyzing the results, you determine that the uncertainty about your unknown concentration determination is too large. a) State 4 ways in which you could decrease the uncertainty. • Measure at more standard concentrations • Make more measurements of the unknown 2 • Improve your technique to reduce s yx • Have more standard concentrations near the ends of the calibration curve • Pick your standard concentrations so that the unknown will be near the center of the calibration line b) Write the important equation(s) that helped you in determining how you can decrease the uncertainty and show how your changes affect the values in this equation. 1−a t 1 n− 2 N + 1 n + (y s − y ) 2 b 2 x i − x i ( ) 2 2 s yx 2 s b 2 yx = ( y i − y ˆ i ) 2 i n − 2 • Measuring more standard concentrations (n) will decrease 1 n , will decrease t, and will decrease s 2 yx ( ) • More standard concentrations near the edges of the calibration curve will increase the x i − x • More measurements of the unknown will decrease 1 N • Picking standard concentrations so the unknown will fall near the center of the calibration curve will decrease y s − y ( ) 2 2 • Improve your technique to reduce s yx to reduce ( ) y i − ˆ y i 4) One area of consideration when performing a hypothesis test is the setting of α- and β-error. a) What are α- and β-errors Describe this in words and in pictures. α-error is the chance of rejecting the null hypothesis when the null hypothesis is true, and β-error is the chance of accepting the null hypothesis when the research hypothesis is true. Values: accept null hypothesis i Values: accept research hypothesis 2 i 2 β α 2
) How is the β-error set The β-error is indirectly set by the α-error (which is directly set) and by the number of measurements. 5) When determining the correct t-value to use in calculating the confidence limit of a number of measurements, only one degree of freedom is lost, while when performing the calculation to determine the correct t-value to use in calculating the confidence limit about a regression line, two degrees of freedom are lost. Why One degree of freedom is lost when calculating the confidence limit about a measurement because the sample mean is calculated. Two degrees of freedom are lost about the regression line are lost because two statistics, the y-intercept and the slope, are calculated. 6) Most of the data sets that you will acquire during your lifetime will only be a sample of a population. Although practical reasons often limit you to small number of points due to time constraints, statistical theory tells us that under most conditions it is better to acquire more points. State why we would want to obtain more data in each of the following cases. Use diagrams or expressions to support your reasoning, if applicable. a) Your data has a Poisson distribution, but you would like to analyze your data by performing an F-test. One of the assumptions of the F-statistic is that the data is obtained from a normal distribution. The F- statistic is not very robust so even data that is obtained from an “almost normal” distribution is suspect. Although the data on which you wish to work follows the Poisson distribution, you can still use the “normal distribution” statistics if the sample is large enough (n>30). The central limit theorem tells us that no matter what the underlying distribution is, the data becomes more normally distributed as more data is taken. b) You are performing a hypothesis test and wish to reduce both the α- and β-error simultaneously. One part of a hypothesis test is to set the α-level, which also sets the β-level. Without taking more data, decreasing the β -level will increase the α -level and vice-versa. Taking more data will narrow the distributions (decrease uncertainty) and therefore limit the values within the distribution that are outside the set threshold(s). c) You are performing a hypothesis test using the t-statistic and wish to accept the research hypothesis at the 95% confidence limit. The research hypothesis is accepted if the calculated t-value is larger than the absolute value of the critical t-value. As the sample size increases, the critical t-value decreases. Also as more data is acquired, the calculated t-value increases as t is proportional to n . Therefore, if there really is a bias, the calculated t-value will eventually become larger than the critical t-value. d) You can calculate the sample mean and wish to decrease the confidence limit of your estimate of the population mean. ts The confidence limit of an estimate of the mean is given as . Therefore, as the number of data n points (n) increases, the confidence limits and uncertainty decrease. 7) Statistics for Analytical <strong>Chem</strong>ists p. 80 <strong>Problem</strong> 6. Answer p. 179. Question: Is this spectroscopic method accurate Statistic: t-test (Case I) Level of significance: 95% confidence limit Hypotheses (General form): H 0 : µ certified = x spectroscopic H 1 : µ certified ≠ x spectroscopic 3
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Page 5: 11) Mermet et al. Analytical Chemis

Answers Problem Set 2 Chem 314A Williamsen Spring 2000

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