Data Set (Carbon has several isotopes, forms with different numbers of neutrons
ID: 3232840 • Letter: D
Question
Data Set
(Carbon has several isotopes, forms with different numbers of neutrons in the nuclei of their atoms. Carbon-12 makes up almost 99% of natural carbon. The abundance of carbon-13 is measured by the ratio of carbon-13 to carbon-12, in parts per thousand more or less than a standard. The minus signs in the data mean that the ratio is smaller in these diamonds than in standard carbon.)
We would like to estimate the mean abundance of both nitrogen and carbon-13 in the population of diamonds represented by this sample.
Examine the data for nitrogen.
This is an opportunity to examine a sample from a single population of diamonds formed in a single event deep in the earth.
Table 20.2 presents data on the nitrogen content (parts per million) and the abundance of carbon-13 in these diamonds.
Data Set
(Carbon has several isotopes, forms with different numbers of neutrons in the nuclei of their atoms. Carbon-12 makes up almost 99% of natural carbon. The abundance of carbon-13 is measured by the ratio of carbon-13 to carbon-12, in parts per thousand more or less than a standard. The minus signs in the data mean that the ratio is smaller in these diamonds than in standard carbon.)
We would like to estimate the mean abundance of both nitrogen and carbon-13 in the population of diamonds represented by this sample.
Examine the data for nitrogen.
Step 1:
Can we use a t confidence interval for mean nitrogen?
Explain your answer. The distribution is very close to Normal, so a t confidence interval can be safely used. The distribution is quite skewed with a strong outlier. Although t procedures are robust, they should not be used if the population being sampled is this heavily skewed. The distribution has a right-hand tail and an outlier, but the t procedure is robust against such features of non-Normality. The distribution is not Normal but it is still symmetric. We should trust a t confidence interval.
Step 2:
Give a 90% confidence interval if you think the result can be trusted. 103.7 to 347.7 ppm. 124.62 to 326.8 ppm. 200.801 to 250.619 ppm. A t confidence interval is not useful for these data. Equation Editor T A BLE 2 o. 2 NITROGEN AND CARBON 13 IN A SAMPLE OF DIAMONDS DIAMOND NITROGEN (PPM) CARBON 13 RATIO DIAMOND NITROGEN (PPM) CARBON 13 RATIO 487 273 2.73 2.78 13 2.33 1430 1.39 14 60 15 69 3.83 4.26 2.04 244 262 1.19 16 2.12 120 2.82 196 17 302 0.84 274 2.87 18 75 41 3.57 3.68 242 2.42 54 3.29 115 3.89 473 3.79 21 4.06 30 3.87 22 65 1.58 1.83 311 98 23 4.03 61 41 24 3.97
Explanation / Answer
Step :1 OPtion C is correct. The distribution has a right-hand tail and an outlier, but the t procedure is robust against such features of non-Normality. The t - distribution takes care of all the possibilites and for small sample size, it is the best parametric test. And Here sample is not normally distributed but population can be assumed normally distributed.
Step:2 Here we have to calculated 90% Confidence interval for nitrogen.
Population mean = sample mean = xbar = Sum of all values/ 24 = 225.71
standard deviation of sample s= 289.00
standard error of the sampling = s/n = 289/24 = 59.00
so 90 % confidence interval = Xbar +- t0.05, 23 (s/n)
= 225.71 +- 1.714 * 59.00 = (124.584, 326.836) Option B is correct.
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