How to Construct a t-Interval for the Slope of  Population Regression Line

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<h2>Transcript</h2>

<p>

All right, Mr. Antonucci here. We’re wrapping up Section 9A by talking about how to 

<strong>calculate and interpret a t-interval for the slope</strong> of a population regression line. 

This is the confidence interval used to estimate the population slope \( \beta \). 

As with any inference procedure, we follow the <strong>four-step process</strong>.

</p>

<h3>Example Context</h3>

<p>

Cars and trucks lose value the more they’re driven. Can we predict the price of a used 

Ford F‑150 Super Crew 4x4 if we know how many miles are on the odometer? 

A random sample of 16 used F‑150s was selected from autotrader.com. 

Miles driven and price (in dollars) were recorded for each truck. 

We’re given the raw data, a scatter plot, a residual plot, computer output, 

and a histogram of the residuals.

</p>

<p>

Our goal: <strong>Construct and interpret a 90% confidence interval</strong> 

for the slope of the population regression line.

</p>

<h3>Step 1: State</h3>

<p>

We will construct a 90% confidence interval for 

\( \beta \), the slope of the population regression line relating 

\( y = \) price to \( x = \) miles driven for used Ford F‑150 Super Crew 4x4s 

listed for sale on autotrader.com.

</p>

<p>

Note the specific context and the correct use of \( \beta \) (population slope), 

not \( b \) (sample slope).

</p>

<h3>Step 2: Plan</h3>

<p>

We will use a <strong>t‑interval for the slope</strong>.  

Before that, we must check the conditions.

</p>

<p><strong>Linear:</strong>  

The scatter plot is roughly linear.  

The residual plot shows no curved pattern; points are randomly scattered.</p>

<p><strong>Normal:</strong>  

The histogram of residuals shows no strong skewness or outliers.  

It doesn’t look perfectly normal—and that’s okay. We are only checking for major issues.</p>

<p><strong>Equal SD:</strong>  

The residuals appear to have a roughly consistent vertical spread across all x-values.</p>

<p><strong>Random:</strong>  

The trucks were selected from a random sample of F‑150s on autotrader.com.</p>

<p><strong>10% Condition:</strong>  

It is reasonable to assume that 16 trucks represent less than 10% 

of all such listings.</p>

<h3>Step 3: Do</h3>

<p>

Degrees of freedom:

\[ df = 16 - 2 = 14 \]

</p>

<p>

The critical value for a 90% confidence interval is:

\[ t^\ast = 1.761 \]

(using inverse t or a table)

</p>

<p>

From the regression output:

<ul>

  <li>Sample slope: \( b = -0.16292 \)</li>

  <li>Standard error of the slope: \( s_b = 0.03096 \)</li>

</ul>

</p>

<p>

Compute the confidence interval:

\[ -0.16292 \pm 1.761(0.03096) \]

</p>

<p>

This gives the interval:

\[ (-0.21744,\; -0.10840) \]

</p>

<h3>Step 4: Conclude</h3>

<p>

We are 90% confident that the interval from 

\( -0.21744 \) to \( -0.10840 \) captures the slope \( \beta \) 

of the population regression line relating price to miles driven 

for used Ford F‑150 Super Crew 4x4s on autotrader.com.

</p>

<p>

In other words, we are 90% confident that on average, 

the price of a used F‑150 decreases by about <strong>11 to 22 cents per mile driven</strong>.

</p>

<h3>Important Testing Advice</h3>

<p>

If you see a list of raw data on an AP Exam, 

<strong>don’t automatically enter it into your calculator</strong>.  

Often the exam provides a regression output that eliminates extra work.

</p>

<p>

Up to this point, we used computer regression output. 

But the TI‑84 can also compute a confidence interval from raw sample data:

</p>

<ol>

  <li>Enter x-values into L1 and y-values into L2.</li>

  <li>Press STAT → TESTS → scroll to <em>LinRegTInt</em>.</li>

  <li>Choose L1 for Xlist, L2 for Ylist, Freq = 1.</li>

  <li>Set the confidence level (e.g., 0.90).</li>

  <li>Select CALCULATE.</li>

</ol>

<p>

The calculator reports the confidence interval, sample slope, degrees of freedom, 

standard deviation of the residuals, y-intercept, \( r^2 \), and \( r \).

</p>

<p><strong>Note:</strong>  

The reported \( s \) value is the <strong>standard deviation of residuals</strong>,  

not the standard error of the slope. Be careful with vocabulary.

</p>

<p><strong>All right guys, that’s it. We’ve wrapped up the section. 

Reach out if you have any questions!</strong></p>