The scatter diagram of the lengths and widths of the petals of a type of flower is football shaped; the correlation is 0.8. The average petal length is 3.1 cm and the SD is 0.25 cm. The average petal width is 1.8 cm and the SD is 0.2 cm.
(a)One of the petals is 0.2 cm wider than another. Regression says that the wider petal is estimated to be ____________ cm longer than the other.
(b)Find the regression estimate of the length of a petal that is 1.5 cm wide.
(c)Of the petals that are 1.5 cm wide, about what percent are less than 2.75 cm long?
(d) The 45th percentile of the petal lengths is (in cm)
(e) The 45th percentile of the lengths of the petals that are 1.8 cm wide is (in cm)
To answer these questions, we need to use the information given about the scatter diagram, correlation, average, and standard deviation of the petal lengths and widths. We'll use the concept of regression and percentiles to find the answers.
(a) The correlation value of 0.8 indicates a strong positive linear relationship between petal widths and lengths. Therefore, we can use regression to estimate the difference in length based on the difference in width.
Since the difference in width is given as 0.2 cm, we can multiply this by the correlation coefficient to estimate the difference in length.
Estimated difference in length = Correlation coefficient * Difference in width
Estimated difference in length = 0.8 * 0.2 = 0.16 cm
So, the wider petal is estimated to be 0.16 cm longer than the other.
(b) To find the regression estimate of the length of a petal that is 1.5 cm wide, we need to use the regression equation derived from the scatter diagram.
Regression equation: Length = Intercept + Slope * Width
To find the Intercept and Slope values, we need to use the average lengths, average widths, and the correlation coefficient.
Intercept = Average length - Slope * Average width
Using the given values:
Average length = 3.1 cm
Average width = 1.8 cm
Correlation coefficient = 0.8
Substituting these values into the equation:
Intercept = 3.1 cm - Slope * 1.8 cm
Now, we need to solve for the Slope.
Slope = Correlation coefficient * (Standard deviation of lengths / Standard deviation of widths)
Slope = 0.8 * (0.25 cm / 0.2 cm)
Slope = 0.8 * 1.25 = 1 cm
Substituting the value of the Slope back into the Intercept equation:
Intercept = 3.1 cm - 1 cm * 1.8 cm
Intercept = 3.1 cm - 1.8 cm
Intercept = 1.3 cm
Now we can substitute the values of Intercept and Slope into the regression equation and find the estimated length of a petal that is 1.5 cm wide:
Length = 1.3 cm + 1 cm * 1.5 cm
Length = 1.3 cm + 1.5 cm
Length = 2.8 cm
Therefore, the regression estimate of the length of a petal that is 1.5 cm wide is 2.8 cm.
(c) To find the percentage of petals that are less than 2.75 cm long among those that are 1.5 cm wide, we need to use the concept of percentiles.
We already know the estimated length of 1.5 cm wide petal, which is 2.8 cm. Now, we need to calculate the cumulative percentage of petals that are less than 2.75 cm long.
Using the regression equation:
Length = 1.3 cm + 1 cm * Width
Substituting Width = 1.5 cm:
Length = 1.3 cm + 1 cm * 1.5 cm
Length = 1.3 cm + 1.5 cm
Length = 2.8 cm
So, the estimated length of a 1.5 cm wide petal is 2.8 cm.
Now, we need to find the cumulative percentage of petals less than 2.75 cm long using the given information about average length and standard deviation.
Z-score = (Value - Average) / Standard deviation
Z-score = (2.75 cm - 3.1 cm) / 0.25 cm
Z-score = -1.4
Using a Z-score table or calculator, we can find that the cumulative percentage associated with a Z-score of -1.4 is approximately 0.0808 or 8.08%.
Therefore, about 8.08% of the petals that are 1.5 cm wide are estimated to be less than 2.75 cm long.
(d) To find the 45th percentile of the petal lengths, we need to use the given average length and standard deviation.
Using a percentile calculator or Z-score table, we can find the corresponding Z-score for the 45th percentile, which is approximately -0.1257.
Z-score = (Value - Average) / Standard deviation
Substituting the known values:
-0.1257 = (Value - 3.1 cm) / 0.25 cm
Solving for the Value:
Value - 3.1 cm = -0.1257 * 0.25 cm
Value - 3.1 cm = -0.031425 cm
Value = 3.1 cm - 0.031425 cm
Value = 3.068575 cm
Therefore, the 45th percentile of the petal lengths is approximately 3.0686 cm.
(e) To find the 45th percentile of the lengths of the petals that are 1.8 cm wide, we need to use the regression equation.
Using the regression equation:
Length = 1.3 cm + 1 cm * Width
Substituting Width = 1.8 cm:
Length = 1.3 cm + 1 cm * 1.8 cm
Length = 1.3 cm + 1.8 cm
Length = 3.1 cm
Since the petal widths are fixed at 1.8 cm, the percentile of the lengths remains the same regardless of the width.
Therefore, the 45th percentile of the lengths of the petals that are 1.8 cm wide is also approximately 3.0686 cm.