The average concentration of ozone in unpolluted tropospheric air is 50 ppbV (that is, fifty molecules of ozone per billion molecules of air). You are trying to determine whether pollution has elevated average tropospheric ozone concentrations above the 50 ppbV “background” level at a particular location. You take five concentration measurements and obtain values of 30, 74, 48, 90, and 85 ppb. Based on other studies of ozone in the troposphere, you assume that these measurements were obtained from a normal distribution. Note in this instance, the null hypothesis should be set up similar to the way I set things up in the lecture for the DeSpoille and Pillage example. (a) What are the mean, standard deviation, and standard error of the mean of measured ozone concentrations? (b) In testing whether the average concentration is elevated above the “background”, what should be yournull hypothesis? Should you use a one- or two-tailed test? (c) Can you reject the null hypothesis at a threshold significance level of a = 0.05? Why or why not? (d) What – approximately – is the actual statistical significance level, p, with which you could reject the null hypothesis given the measured mean, standard deviation, and number of measurements that you have? What is p the probability of? (e) What is the least significant difference, that is, the smallest average increase above background ozone concentrations that would be statistically significant at a = 0.05? (f) What is the least significant number, that is, the smallest number of measurements that would be required to make your measured deviation from background statistically significant at a = 0.05? (g) If pollution has elevated the average ozone concentrations to 75 ppbV, you would want to be sure to detect such a difference in your sampling program. What is the approximate power of your test, with n = 5, to detect deviations of 25 ppbV above background? What is the corresponding value of b? In your own words, explain what b represents. (h) Given that you only have five measurements, and given that they have the variability that you calculated in (a) above, what is the minimum detectable difference, at a statistical significance of a = 0.05 with a power of 90%? (i) If you wanted to be sure to detect deviations of 10 ppbV above background (at a significance level of a = 0.05) with a reliability (i.e., power, or 1 – b) of 90%, what sample size should you be aiming for?
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The average concentration of ozone in unpolluted tropospheric air is 50 ppbV (that is, fifty molecules of ozone per billion molecules of air). You are trying to determine whether pollution has elevated average tropospheric ozone concentrations above the 50 ppbV “background” level at a particular location. You take five concentration measurements and obtain values of 30, 74, 48, 90, and 85 ppb. Based on other studies of ozone in the troposphere, you assume that these measurements were obtained from a
normal distribution . Note in this instance, the null hypothesis should be set up similar to the way I set things up in the lecture for the DeSpoille and Pillage example.-
(a) What are the
mean , standard deviation, and standard error of the mean of measured ozone concentrations? -
(b) In testing whether the average concentration is elevated above the “background”, what should be yournull hypothesis? Should you use a one- or two-tailed test?
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(c) Can you reject the null hypothesis at a threshold significance level of a = 0.05? Why or why not?
-
(d) What – approximately – is the actual statistical significance level, p, with which you could reject the null
hypothesis given the measured mean, standard deviation, and number of measurements that you have?
What is p the probability of?
-
(e) What is the least significant difference, that is, the smallest average increase above background ozone
concentrations that would be statistically significant at a = 0.05?
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(f) What is the least significant number, that is, the smallest number of measurements that would be
required to make your measured deviation from background statistically significant at a = 0.05?
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(g) If pollution has elevated the average ozone concentrations to 75 ppbV, you would want to be sure to detect
such a difference in your sampling program. What is the approximate power of your test, with n = 5, to detect deviations of 25 ppbV above background? What is the corresponding value of b? In your own words, explain what b represents.
-
(h) Given that you only have five measurements, and given that they have the variability that you calculated in (a) above, what is the minimum detectable difference, at a statistical significance of a = 0.05 with a power of 90%?
-
(i) If you wanted to be sure to detect deviations of 10 ppbV above background (at a significance level of a = 0.05) with a reliability (i.e., power, or 1 – b) of 90%, what sample size should you be aiming for?
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