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Weights and heights of turkeys tend to be correlated. For a population of turkeys at a farm, this correlation is found to be 0.64. The average weight is 17 pounds, SD is 5 pounds. The average height is 28 inches and the SD is 8 inches. Weight and height both roughly follow the normal curve. For each part below, answer the question or if not possible, indicate why not. A turkey at the farm which weighs more than 90% of all the turkeys is predicted to be taller than % of them. The average height for turkeys at the 90th percentile for weight is Of the turkeys at the 90th percentile for weight, roughly what percent would you estimate to be taller than 28 inches?

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Answer:

a turkey at the farm which weighs more than 90% of all the turkeys is predicted to be taller than 79.37 % of them.

The  average height for turkeys at the 90th percentile for weight is 34.554

Of the turkeys at the 90th percentile for weight, roughly the percentage that  would  be taller than 28 inches 79.37%

Step-by-step explanation:

The data given for the study can be listed as follows

For a population of turkeys at a farm, the correlation found between the weights and heights of turkeys is r = 0.64

[tex]\overline x[/tex] = 17  (i . e the average weight in pounds)

[tex]S_x[/tex] = 5 ( i . e the standard deviation of the weight in pounds)

[tex]\overline y[/tex] = 28 (i . e the average height in inches)

[tex]S_y[/tex] = 8 ( i . e the standard deviation of the height in inches)

The slope of the regression line can be expressed as :

[tex]\beta_1 = r \times ( \dfrac{S_y}{S_x})[/tex]

[tex]\beta_1 = 0.64 \times ( \dfrac{8}{5})[/tex]

[tex]\beta_1 = 0.64 \times 1.6[/tex]

[tex]\beta_1 = 1.024[/tex]

Similarly the intercept of the regression line can be estimated by using the formula:

[tex]\beta_o = \overline y - \beta_1 \overline x[/tex]

replacing the values, we have:

[tex]\beta_o = 28 -(1.024)(17)[/tex]

[tex]\beta_o = 28 -17.408[/tex]

[tex]\beta_o = 10.592[/tex]

However, the regression line needed for this study can be computed as:

[tex]\hat Y = \beta_o + \beta_1 X[/tex]

[tex]\hat Y = 10.592 + 1.024 X[/tex]

Recall that;

both the weight and height roughly follow the normal curve

Thus, the weight related to 90th percentile can be determined as shown below.

Using the Excel Function at 90th percentile, which can be computed as:

(=Normsinv (0.90) ; we have the desired value of 1.28

∴

[tex]\dfrac{X - \overline x}{s_x } = (Normsinv (0.90))[/tex]

[tex]\dfrac{X - \overline x}{s_x } = 1.28[/tex]

[tex]\dfrac{X - 17}{5} = 1.28[/tex]

[tex]X - 17 = 6.4[/tex]

X = 6.4 + 17

X = 23.4

The predicted height [tex]\hat Y = 10.592 + 1.024 X[/tex]

here; X = 23.4

[tex]\hat Y = 10.592 + 1.024 (23.4)[/tex]

[tex]\hat Y = 10.592 + 23.9616[/tex]

[tex]\hat Y = 34.5536[/tex]

So the probability of predicted height less than 34.5536 can be expressed as:

[tex]P(Y < 34.5536) = P( \dfrac{Y - \overline y }{S_y} < \dfrac{34.5536-28}{8})[/tex]

[tex]P(Y < 34.5536) = P(Z< \dfrac{6.5536}{8})[/tex]

[tex]P(Y < 34.5536) = P(Z< 0.8192)[/tex]

From the Z tables;

P(Y < 34.5536) =0.7937

Thus,  a turkey at the farm which weighs more than 90% of all the turkeys is predicted to be taller than 79.37 % of them.

The  average height for turkeys at the 90th percentile for weight is :

[tex]\hat Y = 10.592 + 1.024 X[/tex]

here; X = 23.4

[tex]\hat Y = 10.592 + 1.024 (23.4)[/tex]

[tex]\hat Y = 10.592 + 23.962[/tex]

[tex]\mathbf{\hat Y = 34.554}[/tex]

Of the turkeys at the 90th percentile for weight, roughly what percent would you estimate to be taller than 28 inches?

This implies that :

P(Y >28) = 1 - P (Y< 28)

[tex]P(Y >28) = 1 - P( Z < \dfrac{28 - 34.554}{8})[/tex]

[tex]P(Y >28) = 1 - P( Z < \dfrac{-6.554}{8})[/tex]

[tex]P(Y >28) = 1 - P( Z < -0.8193)[/tex]

From the Z tables,

[tex]P(Y >28) = 1 - 0.2063[/tex]

[tex]\mathbf{P(Y >28) = 0.7937}[/tex]

= 79.37%