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Archaeology: Ireland Inorganic phosphorous is a naturally occurring element in all plants and animals, with concentrations increasing progressively up the food chain (fruit \(<\) vegetables \(<\) cereals \(<\) nuts \(<\) corpse). Geochemical surveys take soil samples to determine phosphorous content (in ppm, parts per million). A high phosphorous content may or may not indicate an ancient burial site, food storage site, or even a garbage dump. The Hill of Tara is a very important archaeological site in Ireland. It is by legend the seat of Ireland's ancient high kings (Reference: Tara, An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Independent random samples from two regions in Tara gave the following phosphorous measurements (in ppm). Assume the population distributions of phosphorous are mound-shaped and symmetric for these two regions. $$ \begin{aligned} &\text { Region I: } x_{1} ; n_{1}=12\\\ &\begin{array}{llllll} 540 & 810 & 790 & 790 & 340 & 800 \\ 890 & 860 & 820 & 640 & 970 & 720 \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Region II: } x_{2} ; n_{2}=16\\\ &\begin{array}{llllllll} 750 & 870 & 700 & 810 & 965 & 350 & 895 & 850 \\ 635 & 955 & 710 & 890 & 520 & 650 & 280 & 993 \end{array} \end{aligned} $$ (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x}_{1} \approx 747.5, s_{1} \approx 170.4, \bar{x}_{2} \approx 738.9\), and \(s_{2} \approx 212.1 .\) (b) Let \(\mu_{1}\) be the population mean for \(x_{1}\) and let \(\mu_{2}\) be the population mean for \(x_{2} .\) Find a \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). (c) Interpretation Explain what the confidence interval means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the \(90 \%\) level of confidence, is one region more interesting than the other from a geochemical perspective? (d) Check Requirements Which distribution (standard normal or Student's \(t\). did you use? Why?

Step by step solution

01

Verify Given Mean and Standard Deviation

Using a calculator, calculate the mean and standard deviation for each region. For Region I, sum up all values and divide by 12 to get the mean \( \bar{x}_1 \). The handheld calculation yields \( \bar{x}_1 \approx 747.5 \). For the standard deviation \( s_1 \), use the formula for standard deviation. Similarly, do the calculation for Region II to confirm \( \bar{x}_2 \approx 738.9 \) and \( s_2 \approx 212.1 \). The calculations match the given information, verifying the data.

02

Calculate the Confidence Interval

Use the formula for the confidence interval of the difference between two means: \[ CI = (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]where \( t_{\alpha/2} \) is the value from the t-distribution table for 90% confidence with degrees of freedom based on the smaller sample size minus one, which in this case approximately equals 2.074. Calculate:\[ (747.5 - 738.9) \pm 2.074 \cdot \sqrt{\frac{170.4^2}{12} + \frac{212.1^2}{16}} \]By solving it, the confidence interval approximately equals \( (-92.08, 111.88) \).

03

Interpret the Confidence Interval

The confidence interval \((-92.08, 111.88)\) includes both negative and positive numbers. This indicates that there is no significant difference between the means of the two regions at the 90% confidence level. The phosphorus levels could be higher in Region I or Region II, or they could be the same. This shows neither region is geochemically more interesting at this confidence level.

04

Check Requirements

Since the population distributions are assumed to be symmetric and mound-shaped (normal-like) and the sample sizes are not very large, we use the Student's t-distribution. The t-distribution is more appropriate in cases where the sample sizes are small and the population standard deviation is unknown.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

A confidence interval provides a range of values that estimates an unknown population parameter. In the context of the archaeological site at Tara, Ireland, the confidence interval is used to estimate the difference in mean phosphorous levels between two regions. This range gives us a 90% confidence that the true difference in phosphorous levels falls within this interval.
This particular confidence interval is calculated using the means and standard deviations of phosphorous measurements from samples collected in each region. The interval is \( -92.08, 111.88 \), encompassing negative, positive, and zero.
This means that, at a 90% confidence level, either region could have higher or lower phosphorous levels, or they could be similar. It indicates uncertainty over which region is more geochemically interesting based on phosphorous content alone.

Standard Deviation

Standard deviation is a measure of how spread out the values in a data set are. For the two regions at Tara, we calculate the standard deviation of phosphorous levels to understand the variability within each group's measurements.
The formulas for standard deviation help determine how tightly the data points cluster around the mean. In Region I, the standard deviation \( s_1 \) is approximately 170.4, while in Region II, it is 212.1. These values indicate the degree of dispersion around the mean phosphorous level for each region. A larger standard deviation reflects more variation in phosphorous content, while a smaller one suggests the values are closer to the mean.
Understanding standard deviation is crucial when comparing two or more data sets, as it directly affects the width of the confidence interval and our interpretation of the data.

Mean

The mean is a basic measure of the central tendency of a data set. It is calculated by adding up all the measurements and dividing by the number of observations. In the Tara archeological survey, the mean phosphorous content for Region I is approximately 747.5 ppm, while for Region II, it is 738.9 ppm.
These means represent the average phosphorous levels for each region, offering a snapshot of typical values that can be expected. Means are used in statistics to summarize data sets and form the basis for further calculations, such as constructing confidence intervals.
Keep in mind that while means provide valuable insights into data, they can be influenced by extreme values (outliers), which makes understanding the dataset's distribution and standard deviation equally important.

Student's t-distribution

The Student's t-distribution is utilized when dealing with small sample sizes and unknown population standard deviations. It is similar to the normal distribution but has fatter tails, which provide a more conservative estimate. This distribution is used to calculate the confidence interval for the mean phosphorous level difference between the two regions at Tara.
Given that the sample sizes are relatively small (12 for Region I and 16 for Region II) and the population distributions are symmetric and mound-shaped, the use of the t-distribution is appropriate. For a 90% confidence level, the critical value \( t_{\alpha/2} \) is approximately 2.074 in this case.
Using the t-distribution ensures that even with smaller samples, the confidence interval formed is reliable and accounts for the increased variability, providing a better reflection of uncertainty in small sample scenarios.