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2 years ago

Carbon-14 has a half-life of approximately 5,730 years. This exponential decay can be modeled with the function N(t) = N0.

Mathematics

1 answer:

kvasek [131]2 years ago

4 0

$\textit{Amount for Exponential Decay using Half-Life} \\\\ A=P\left( \frac{1}{2} \right)^{\frac{t}{h}}\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &200\\ t=\textit{elapsed time}\dotfill &14325\\ h=\textit{half-life}\dotfill &5730 \end{cases} \\\\\\ A=200\left( \frac{1}{2} \right)^{\frac{14325}{5730}}\implies A=200\left( \frac{1}{2} \right)^{\frac{5}{2}}\implies A\approx 35.36$

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Answer:

a) $\bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75$

b) The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean

Step-by-step explanation:

Part a

The best point of estimate for the population mean is the sample mean given by:

$\bar X = \frac{\sum_{i=1}^n X_i}{n}$

Since is an unbiased estimator $E(\bar X) = \mu$

Data given: 2 , 2 , 1 , 3 , 1 , 0 , 4 , 1

So for this case the sample mean would be:

$\bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75$

Part b

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X$ represent the sample mean for the sample