Intervals on Means

\[ \begin{aligned} \text{Point estimate} & = \bar x \\ \\ \text{Margin of error} & = t_{\alpha/2} \ \frac{s}{\sqrt n} \\ \end{aligned} \]

\[\bar x \pm t_{\alpha/2} \ \frac{s}{\sqrt n}\]

Intervals on Means

Intervals on Means

\[ \begin{aligned} \bar{x} = 460.62 \\ s_x = 205.77 \\ n = 4679 \\ \end{aligned} \]

Intervals on Means

Intervals on Means

\[ \begin{aligned} \bar{x} = 460.62 \\ s_x = 165.38 \\ n = 64 \\ \end{aligned} \]

Proportions

Why have I never really talked about these?

• Proportions are important

  • Difficult for newer stats students

  • Easy once you know a little bit

• All the rules are “different” (but not really)

Intervals on Proportions

\[\text{Point estimate} \pm \text{Margin of error}\]

\[ \hat{p} = \frac{\text{TDS} > 500 \text{ mg/L}}{\text{Total n of TDS}} \]

\[ \hat{p} = \frac{1267}{4679} \]

Intervals on Proportions

Intervals on Proportions

\[ \begin{aligned} \text{Point estimate} & = \hat{p} \\ \\ \text{Margin of error} & = z_{\alpha/2} \ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\ \end{aligned} \]

\[ \hat{p} \pm z_{\alpha/2} \ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

Intervals on Proportions

\[ \begin{aligned} \hat{p} = 0.271 \\ n = 4679 \\ \end{aligned} \]

Intervals on Proportions

Intervals on Proportions

\[ \begin{aligned} \hat{p} = 0.297 \\ n = 64 \\ \end{aligned} \]

• To use a \(z\)-table we need to meet an assumption:

\[ \hat{p} \sim N(\mu_{\hat{p}},\sigma^2_{\hat{p}}) \]

• This is true if either of the following are satisfied:

\[ n\hat{p} \ge 10 \text{ or } n(1-\hat{p}) \ge 10 \]

Intervals on Proportions

\[ \begin{aligned} \hat{p} = 0.297 \\ n = 64 \\ \end{aligned} \]

Regression

Statistical Inference

• What does our model tell us about the data?

  • About the process? Science as a whole?

• Is our model and its assumptions reasonable?

• What decisions can we make given the results we have?

Intervals in Regression

\[ \begin{aligned} \text{MSE}\ =& \ \frac{\text{RSS}}{n-2} \\ SE(\hat\beta_0)= & \ \sqrt{\text{MSE}\left( \frac{1}{n}+\frac{\bar x^2}{\sum_{i=1}^n(x_i+\bar x)^2}\right)}\\ \\ SE(\hat \beta_1)= & \ \sqrt{\frac{\text{MSE}}{\sum_{i=1}^n(x_i-\bar x)^2}}\\ \end{aligned} \]

Intervals in Regression

\[ \begin{aligned} CI_{\hat \beta_0}= & \ \hat \beta_0 \pm t^* \times SE(\hat \beta_0) \\ \\ CI_{\hat \beta_1}= & \ \hat \beta_1 \pm t^* \times SE(\hat \beta_1) \\ \end{aligned} \]

Intervals in Regression

Call:
lm(formula = Y ~ X)

Residuals:
    1     2     3 
-0.54  1.08 -0.54 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.3400     2.0205  -0.168    0.894
X             1.0400     0.9353   1.112    0.466

Residual standard error: 1.323 on 1 degrees of freedom
Multiple R-squared:  0.5529,    Adjusted R-squared:  0.1057 
F-statistic: 1.236 on 1 and 1 DF,  p-value: 0.4663

Intervals in Regression

\[ CI_{\hat \beta_0}= \begin{cases} -0.34- 12.706\times 2.0205 \\ -0.34 + 12.706 \times 2.0205 \end{cases} \\ \]

\[ CI_{\hat \beta_1}= \begin{cases} 1.04- 12.706\times 0.9353\\ 1.04 + 12.706 \times 0.9353 \end{cases} \]

Intervals in Regression

\[ \begin{aligned} CI_{\hat \beta_0}= & \ (-26.0125,25.3325)\\ \\ CI_{\hat \beta_1}= & \ (-10.8439,12.9239)\\ \end{aligned} \]

• What do we make of these?

Inference in Observation

Inference in Observation

Call:
lm(formula = Y ~ X)

Residuals:
    Min      1Q  Median      3Q     Max 
-180.75 -101.87  -35.19   51.47  798.13 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 63396.17   35849.36   1.768   0.0819 .
X             -31.12      17.73  -1.755   0.0841 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 162.7 on 62 degrees of freedom
Multiple R-squared:  0.04735,   Adjusted R-squared:  0.03198 
F-statistic: 3.081 on 1 and 62 DF,  p-value: 0.08414

Inference in Observation

\[ 63396.17 \pm 1.96 \times 35849 = \]

\[ -31.12 \pm 1.96 \times 17.73 \]

                  2.5 %       97.5 %
(Intercept) -8265.67415 1.350580e+05
X             -66.55953 4.318761e+00

Inference in Observation