Problem 3. Remember that cesium-134 atoms are radioactive, and the time until an

Question

Problem 3. Remember that cesium-134 atoms are radioactive, and the time until an atom decays has an exponential distribution with 0.33485yr-1 (corresponding to a half-life of 2.07 years). You have a collection of 100 radioactive atoms, but you don't know whether they are cesium-134 or something else. So, you measure the decay times of 52 of the atoms, and compare their average decay time to that of cesium-134. You suspect that they are cesium-134, and will continue to believe this unless you have strong evidence to the contrary (you want to perform the test with a significance level of 0.10). Here are the results from your test (decay times measured in years; yes, you are very patient.)

Explanation / Answer

(a) H0 : The atoms belong to cesium - 134. Average Decay time is equal to standard decay time of cesium - 134. =0.33485 year-1

Ha : The atoms doesn't belong to cesium - 134. Average Decay time is equal to standard decay time of cesium - 134.   0.33485 year-1

(b) Here we will calculate 1/time period in excel and then we will take average of all these values.

Average   = 0.4088 year-1

standard deviation of sample s = 0.1552 year-1

standard error of sample mean = s/sqrt(n) = 0.1552/ sqrt(52) = 0.02152

Here test statistic

t = (0.4088 - 0.33485)/ 0.02152 = 3.4363

so here dF = 51, alpha = 0.05

so tcritical = t51,0.05 = 2.0076

here t > tcritical so we shall reject the null hypothesis and can say the atoms don't belong to cesium atoms.

Time Rate constant 1.41 0.7092 1.45 0.6897 1.55 0.6452 1.6 0.6250 1.6 0.6250 1.67 0.5988 1.69 0.5917 1.71 0.5848 1.75 0.5714 1.78 0.5618 1.78 0.5618 1.81 0.5525 1.82 0.5495 1.83 0.5464 1.83 0.5464 1.9 0.5263 1.92 0.5208 1.94 0.5155 2 0.5000 2 0.5000 2.08 0.4808 2.08 0.4808 2.1 0.4762 2.15 0.4651 2.35 0.4255 2.75 0.3636 2.75 0.3636 2.8 0.3571 2.83 0.3534 3 0.3333 3 0.3333 3.16 0.3165 3.17 0.3155 3.17 0.3155 3.25 0.3077 3.35 0.2985 3.39 0.2950 3.5 0.2857 3.5 0.2857 3.5 0.2857 3.67 0.2725 4 0.2500 4 0.2500 4.13 0.2421 4.17 0.2398 4.75 0.2105 5 0.2000 5.1 0.1961 5.18 0.1931 5.44 0.1838 5.5 0.1818 5.7 0.1754 Average 0.4088 Std. Dev. 0.1552

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