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4-56.

In problem 4-25 you looked at the data for a study conducted on a vitamin supplement that claims to shorten the length of the common cold. The data is shown in the table below:

 Number of months taking supplement $0.5$ $2.5$ $1$ $2$ $0.5$ $1$ $2$ $1$ $1.5$ $2.5$ Number of days cold lasted $4.5$ $1.6$ $3$ $1.8$ $5$ $4.2$ $2.4$ $3.6$ $3.3$ $1.4$
1. You previously created a linear model for this data by sketching a line of best fit. Now create a model that is consistent with your classmates by determining the LSRL. Sketch the graph and the LSRL.

Review the LSRL procedure in the previous lesson.

2. Draw the upper and lower boundary lines on the graph following the process you used in problem 4-24. What is the equation of the upper boundary line? Of the lower boundary line?

Answer: $y = 6.16 − 1.58x$ and $y = 4.58 − 1.58x$, based on a maximum residual of $−0.79$

3. Based on the upper and lower boundary lines of your model, what do you predict is the length of a cold for a person who has taken the supplement for three months? Consider the precision of the data and use an appropriate number of decimal places in your response.

Since the number of days the cold lasted is decreasing as the number of months taking the supplement increases,
it would seem the length of the cold should be between $1$ and $0.5$ days.

4. How long do you predict a cold will last for a person who has taken no supplement? Interpret the y-intercept in context.

Since the $y$-intercept is around $5.3$, the number of days a cold should last for a person who hasn't taken the supplement should be around $5.3$ days.

5. How long do you predict a cold will last for a person who has taken six months of supplements?

Negative values in this case do not make sense. Statistical models often cannot be extrapolated far beyond the edges of the data.

6. If you have a cold, would you prefer a negative or positive residual?

• If you have a cold, your goal would be to decrease the number of days the cold lasts. Would a positive or negative residual represent a decrease in the number of days that a cold lasts?