Probability & Statistics Honors Unit Test #4

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 1 In the mid-1990s, data concerning the size (carets) and price (dollars) for small diamonds were used to establish the following model: $\stackrel{^}{\text{price}}=-259.63+3721.02\left(\text{size}\right)$ The residual for a particular diamond was $410.1586. If the size of that diamond was 0.18 carets, what was the actual price? (4) The predicted price for that diamond was$ $-259.63+3721.02\left(0.18\right)\approx 410.1536$; since $\text{residual}=\text{observed}-\text{predicted}$, $410.1586=\text{observed}-410.1536⇒\text{observed}=410.1586+410.1536=820.3122$ dollars.

 2 Researchers at Brigham Young University collected many body measurements from participants in a study. A graph of some of the data is shown above. Identify and classify any unusual points in the plot. (3)

The point near $\left(125,130\right)$ has leverage; the point near $\left(149,135\right)$ is influential.

 3 An analyst obtained data from a random sample of household electric bills, and found the following relationship between the amount of the bill (dollars) and the average temperature (°F) for the month: $\stackrel{^}{\text{bill}}=185.22-1.5674\left(\text{temperature}\right)$ Use this model to predict the bill in a month where the average temperature was 65°F. (3)

$185.22-1.5674\left(65\right)=83.339$ dollars.

A paper in the American Journal of Physical Anthropology provided some measurements of several skeletons—the length of the metacarpal bones in the palm (cm), and the height (cm). A portion of the collected data is given below.

 Palm 45 51 39 41 48 49 46 43 47 Height 171 178 157 163 172 183 173 175 173

Use this information for all other questions on this test.

 4 Construct a model useful for predicting the height of a skeleton from the length of the metacarpal bones. (4)

$\stackrel{^}{\text{height}}=94.428+1.7\left(\text{palm}\right)$

 5 Interpret the slope of your equation from [4]. (4)

For each additional centimeter of metacarpal length in the palm, the model predicts an average increase of 1.7 cm in height.

 6 Interpret the y-intercept of your equation from [4]. (4)

When the length of the metacarpals in the palm are zero cm, the model predicts an average height of 94.428 cm.

 7 Calculate and interpret the coefficient of determination for these data. (4)

${r}^{2}=0.7327$; about 73.27% of the variation in height can be explained by a linear relationship with metacarpal length of bones in the palm.

 8 Predict the height of a skeleton with 44 cm metacarpal bones. (3)

$94.428+1.7\left(44\right)\approx 169.2116$ cm

 9 Would it be appropriate to use this model to predict the metacarpal length for a skeleton that is 185 cm tall? (3)

No—this model can only be used to predict height.

 10 Construct a residual plot for these data. (5)

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