Probability & Statistics Honors Unit Test #4

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 1 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: $\stackrel{^}{\text{bill}}=185.22-1.5674\left(\text{temperature}\right)$ The residual for a particular month was -\$44.7909. If the average temperature that month was 32.5°F, what was the amount of the electric bill? (4)

When the average temperature is 32.5, the predicted bill is 134.2795. The residual is $\text{observed}-\text{expected}$, so $-44.7909=\text{observed}-134.2795⇒\text{observed}=134.2795-44.7909=89.4886$ dollars.

 2 Researchers in California 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 circled point near $\left(75,110\right)$ has leverage and appears to be an outlier (making it influential); the circled point near $\left(72,129\right)$ is an outlier.

 3 In the mid-1990s, data concerning the size (carets) and price (dollars) for small diamonds were used to establish the following model: $\stackrel{^}{\text{price}}=-259.63+3721.02\left(\text{size}\right)$ Use this model to predict the price of a 0.3 caret diamond. (3)

$\stackrel{^}{\text{price}}=-259.63+3721.02\left(0.3\right)\approx 856.676$ dollars.

Statistics from a sample of Major League Baseball players were collected during the 1992 season. A portion of the collected data is given below.

 Runs 74 32 87 16 102 6 0 39 4 65 58 41 21 46 88 Hits 129 51 169 26 163 25 3 64 8 144 144 55 56 106 188

Use this information for all other questions on this test.

 4 Construct a model useful for predicting the number of runs a player had from the number of hits. (4)

The least squares regression model is $\stackrel{^}{\text{runs}}=0.9974+0.4989\left(\text{hits}\right)$.

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

For each additional hit a player has, the model predicts an average increase of 0.4989 runs.

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

When a player has zero hits, the model predicts an average of 0.9974 runs.

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

${r}^{2}=0.92003$; about 92% of the variation in the number of runs can be explained by a linear relationship with the number of hits.

 8 Predict the number of runs for a player with 75 hits. (3)

$\stackrel{^}{\text{runs}}=0.9974+0.4989\left(75\right)\approx 38.41507$

 9 Would it be appropriate to use this model to predict the number of hits for a player with 110 runs? (3)

No. This model can only be used to predict runs from the number of hits.

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

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