Doctors are interested in the relationship between the...

Doctors are interested in the relationship between the dosage of a medicine and the time required for a patient's recovery. The following table shows, for a sample of 10 patients, dosage levels (in grams) and recovery times (in hours). These patients have similar characteristics except for medicine dosages. $$ \begin{array}{lrrrrrrrrrr} \hline \text { Dosage level } & 1.2 & 1.3 & 1.0 & 1.4 & 1.5 & 1.8 & 1.2 & 1.3 & 1.4 & 1.3 \\ \hline \text { Recovery time } & 25 & 28 & 40 & 38 & 10 & 9 & 27 & 30 & 16 & 18 \\ \hline \end{array} $$ a. Estimate the linear regression of recovery time on dosage level. b. Find and interpret a $90 \%$ confidence interval for the slope of the population regression line. c. Would the sample regression derived in part a be useful in predicting recovery time for a patient given 2.5 grams of this drug? Explain your answer.

Doctors are interested in the relationship between the dosage of a medicine and the time required for a patient's recovery. The following table shows, for a sample of 10 patients, dosage levels (in grams) and recovery times (in hours). These patients have similar characteristics except for medicine dosages.
$$
\begin{array}{lrrrrrrrrrr}
\hline \text { Dosage level } & 1.2 & 1.3 & 1.0 & 1.4 & 1.5 & 1.8 & 1.2 & 1.3 & 1.4 & 1.3 \\
\hline \text { Recovery time } & 25 & 28 & 40 & 38 & 10 & 9 & 27 & 30 & 16 & 18 \\
\hline
\end{array}
$$
a. Estimate the linear regression of recovery time on dosage level.
b. Find and interpret a $90 \%$ confidence interval for the slope of the population regression line.
c. Would the sample regression derived in part a be useful in predicting recovery time for a patient given 2.5 grams of this drug? Explain your answer.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between two quantitative variables by fitting a linear equation to observed data. In simple linear regression, one variable is treated as the predictor (independent variable) and the other as the outcome (dependent variable). The method estimates an intercept and a slope, where the slope shows the expected change in the outcome for a one-unit change in the predictor, and the intercept represents the expected outcome when the predictor is zero. The technique is grounded in minimizing the sum of squared differences between observed and predicted values, and it rests on assumptions such as linearity, independence, homoscedasticity, and normally distributed errors.

Confidence Interval for Regression Slope

A confidence interval for the regression slope is a range of values, derived from the sample data, that is likely to contain the true slope of the population regression line at a given level of confidence (e.g., 90%). This interval provides a measure of the reliability of the estimated slope. It is calculated using the standard error of the slope estimate and a critical value from the t-distribution corresponding to the desired confidence level and degrees of freedom. A confidence interval that does not include zero suggests a statistically significant relationship between the predictor and the outcome.

Prediction and Extrapolation

Prediction in regression involves using the estimated model to forecast the outcome variable for given values of the predictor variable. However, it is important to distinguish between interpolation and extrapolation. Interpolation refers to prediction within the range of the observed data, where the model is generally reliable. Extrapolation, on the other hand, involves predicting values outside the range of the observed data, and such predictions can be unreliable because the linear relationship established by the model may not hold outside the data range. Thus, applying the model to predict recovery time for a dosage that is not represented within the sample can be problematic.

Transcript

00:01 So we have 18 pieces of data here, and we're not going to write all these naturally down.

00:06 So we have a sample size of 18 sets of data.

00:09 And we have x data and y data.

00:11 And the x was standing for the number of units of the drug that was given.

00:15 And this first person had five, and the second had 53, the time of recovery, i believe it was, days for recovery from this.

00:26 And so we want to find our covariance and our correlation coefficient.

00:32 And so we do need to find what our x bar and y bar is.

00:36 So if i do my stat, calculate two variable statistics, i can get what those values are very nicely.

00:43 And we find that the mean of the x's comes out to be 11 .611 .1.

00:50 And the standard deviation of those xes comes out to be 5 .64, we'll call it.

00:57 It 8.

00:57 And for our y's, our y bar is 55 .2 repeating, so let us go 55 .222.

01:05 And our standard deviation of y is 5 .88, and we'll call that 7, round it off to three decimal places.

01:14 So next step is we need to find what the deviation is of x, or x is minus our x bar.

01:23 And we'll put that in list.

01:24 This is in list 1, list 2, and this will be in list 3.

01:28 And then our list 4 will have our y values minus the y bar.

01:35 And so i'll just write the first cell down.

01:37 So i'll go back to my spreadsheet.

01:39 I'll get up into my list 3 and have list 1 minus my x bar, which i can get from variables, statistics, and get the x bar.

01:50 And we end up that first cell becomes negative 6 .6 .1.

01:55 And then now this value, which we could do these all longhand too, would just be kind of tedious.

02:01 And so i take my list two minus my y bar, which again, i can find from variable statistics.

02:09 And that y bar is right there.

02:11 And that first cell is negative 2 .2222.

02:15 Now we need to multiply those two together...

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