## Related questions with answers

In the earlier exercise, data were given on the adjusted gross income $x$ and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation $\hat{y}= 4.68 + .16x$, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of 52,500 dollars is 13,080 dollars. $\newline$ Develop a $95\%$ confidence interval for the mean amount of total itemized deductions for all taxpayers with an adjusted gross income of 52,500 dollar.

Solutions

VerifiedThe mentioned exercise gives $n=7$ data points, which are fitted to a linear model. This exercise focuses on point $y^*$ at $x^*=52.5$. The formula for the $(1-\alpha)\cdot 100\%$ confidence interval for $E(y^*)$ is:

$\hat{y^*}\pm t_{\alpha/2}\cdot s_{\hat{y^*}}.$

Estimation $\hat{y^*}=13.08$ from the regression equation is given.

Given:

$\begin{align*} n&=\text{Sample size}=7 \\ x_p&=\text{Given $x$-value}=52.5 \\ \hat{y}_p&=\text{Predicted $y$-value}=13.08 \\ c&=\text{Confidence level}=95\%=0.95 \\ \sum (x_i-\overline{x})^2&=7648 \\ \overline{x}&=\frac{\sum x}{n}=57 \\ &\color{#4257b2}\text{Result previous exercise} \\ \hat{y}&=4.68+0.16x \end{align*}$

In this exercise, we determine a confidence interval for the mean of $y$ at $x_p=52.5$.

*How can the confidence interval be derived?*

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