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The article "The Incorporation of Uranium and Silver by Hydrothermally Synthesized Galena" (Econ. Geology, 1964: 1003-1024) reports on the determination of silver content of galena crystals grown in a closed hydrothermal system over a range of temperature. With \(x=\) crystallization temperature in \({ }^{\circ} \mathrm{C}\) and \(y=\mathrm{Ag}_{2} \mathrm{~S}\) in mol\%, the data follows: $$ \begin{array}{l|ccccccccccccc} x & 398 & 292 & 352 & 575 & 568 & 450 & 550 & 408 & 484 & 350 & 503 & 600 & 600 \\ \hline y & .15 & .05 & .23 & .43 & .23 & .40 & .44 & .44 & .45 & .09 & .59 & .63 & .60 \end{array} $$ from which \(\sum x_{i}=6130, \sum x_{i}^{2}=3,022,050, \sum y_{i}=4.73\), \(\sum y_{i}^{2}=2.1785, \sum x_{i} y_{i}=2418.74, \hat{\beta}_{1}=.00143, \hat{\beta}_{0}=-.311\), and \(s=.131\). a. Estimate true average silver content when temperature is \(500^{\circ} \mathrm{C}\) using a \(95 \%\) confidence interval. b. How would the width of a \(95 \%\) CI for true average silver content when temperature is \(400^{\circ} \mathrm{C}\) compare to the width of the interval in part (a)? Answer without computing this new interval. c. Calculate a \(95 \%\) CI for the true average change in silver content associated with a \(1^{\circ} \mathrm{C}\) increase in temperature. d. Suppose it had previously been believed that when crystallization temperature was \(400^{\circ} \mathrm{C}\), true average silver content would be .25. Carry out a test at significance level .05 to decide whether the sample data contradicts this prior belief.

Step by step solution

01

Understanding the Linear Regression Equation

The given data can be analyzed using a simple linear regression model: \[ y = \hat{\beta}_0 + \hat{\beta}_1 x \] where \( y \) is the silver content, \( x \) is the temperature, \( \hat{\beta}_0 = -0.311 \) is the intercept, and \( \hat{\beta}_1 = 0.00143 \) is the slope of the regression line.

02

Estimate True Average Silver Content at 500°C

To estimate the true average silver content at \( x = 500 \), substitute this value into the regression equation:\[ \hat{y} = -0.311 + 0.00143 \times 500 = 0.354 \]Calculate the estimated true average and then construct a 95% confidence interval around it. The formula for the CI is:\[ \hat{y} \pm t_{\alpha/2, n-2} \times s \sqrt{\frac{1}{n} + \frac{(x - \bar{x})^2}{\sum (x_i - \bar{x})^2}} \]Where \( s = 0.131 \), \( n \) is the number of samples, and \( \bar{x} = \frac{\sum x_i}{n} \). Use the given \( \sum x_i = 6130 \) and \( n = 13 \) to find \( \bar{x} \) and apply to the formula.

03

Compare Widths of 95% CIs at 400°C and 500°C

While absolute computations aren't necessary, the width of the confidence interval depends on the term \( \frac{(x - \bar{x})^2}{\sum (x_i - \bar{x})^2} \). A temperature of 400°C is closer to the average temperature \( \bar{x} \) than 500°C, leading to a smaller numerator and thus a narrower confidence interval.

04

Calculate 95% CI for True Average Change per °C

The slope \( \hat{\beta}_1 = 0.00143 \) represents the average change in \( y \) per 1°C change in \( x \). To create a 95% confidence interval for \( \hat{\beta}_1 \), use:\[ \hat{\beta}_1 \pm t_{\alpha/2, n-2} \times se(\hat{\beta}_1) \]where \( se(\hat{\beta}_1) = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}} \).Substitute the given values and \( \alpha = 0.05 \).

05

Hypothesis Test for Belief at 400°C

Conduct a hypothesis test for the prior belief \( H_0: \mu_y = 0.25 \) at 400°C. Calculate the test statistic:\[ t = \frac{\hat{y} - 0.25}{se(\hat{y})} \]where \( \hat{y} = -0.311 + 0.00143 \times 400 \). Use a two-tailed t-test with critical values from a t-distribution for \( n-2 \) degrees of freedom. Reject \( H_0 \) if the statistic falls outside the critical t-values.

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Key Concepts

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

Confidence Interval

Confidence intervals are vital for estimating the range within which a true parameter value lies, based on sample data. When we talk about the confidence interval for the average silver content in galena crystals at a certain temperature, we refer to the range we expect the true average value to fall into, with 95% certainty.

The formula for calculating a confidence interval in linear regression involves several components: the estimated coefficient (\(\hat{y}\)), the standard deviation of residuals (\(s\)), and the distribution of the sample means around their average (\(\bar{x}\)). The formula is given by:

• \(\hat{y} \pm t_{\alpha/2, n-2} \times s \sqrt{\frac{1}{n} + \frac{(x - \bar{x})^2}{\sum (x_i - \bar{x})^2}}\)

This formula considers the variability in our estimates and accounts for the spread of the data.

A narrower confidence interval indicates a more precise estimate, while a wider interval reflects greater uncertainty. For instance, in the original problem, the confidence interval is wider at 500°C than at 400°C because 400°C is closer to the mean crystallization temperature. Thus, confidence intervals help us understand the precision of our estimates and compare different scenarios.

Hypothesis Testing

Hypothesis testing is a method used to make inferences about population parameters based on sample statistics. In the context of linear regression, we often test hypotheses about the regression coefficients or about predicted values at specific values of the independent variable.

For instance, if there is a prior belief that the average silver content at 400°C is 0.25, a hypothesis test can help determine whether the data supports or contradicts this belief. We set up a null hypothesis (\(H_0\)) that reflects the prior belief: \(\mu_y = 0.25\), and an alternative hypothesis (\(H_a\)) that suggests a different value. A t-test statistic is then calculated to evaluate the hypothesis:

• \(t = \frac{\hat{y} - 0.25}{se(\hat{y})}\)

where \(\hat{y}\) is the predicted value at 400°C, and \(se(\hat{y})\) is the standard error of the predicted value.
Decisions are made based on the t-statistic: if it falls outside a critical range (determined by the significance level \(\alpha\)), the null hypothesis is rejected, suggesting evidence for the alternative hypothesis. This process helps in verifying or challenging established beliefs with statistical evidence.

Regression Slope

The regression slope (\(\hat{\beta}_1\)) in a linear model tells us how much the dependent variable (\(y\)) is expected to change for a one-unit increase in the independent variable (\(x\)). It quantifies the relationship between variables, indicating the strength and direction of their linear association.

In the provided problem, the slope of the regression line is 0.00143, suggesting that for each 1°C increase in temperature, the silver content increases by 0.00143 mol%. To evaluate the accuracy of this slope, a confidence interval is calculated:

• \(\hat{\beta}_1 \pm t_{\alpha/2, n-2} \times se(\hat{\beta}_1)\)

Here, \(se(\hat{\beta}_1)\) is the standard error, which accounts for the variability of \(\hat{\beta}_1\).
Understanding the regression slope is crucial because it offers insights into how the response variable behaves concerning changes in the predictor, confirming or disproving hypotheses about data trends. A significant slope indicates a meaningful relationship, critical for making predictive and interpretative decisions. This guides researchers and practitioners in applying statistical analysis to real-world scenarios.