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.
1. 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.
2. Find the regression estimate of the length of a petal that is 1.5 cm wide.
3. Of the petals that are 1.5 cm wide, about what percent are less than 2.75 cm long?
To answer these questions, we need to use the information provided about the scatter diagram, correlation coefficient, and the standard deviations of petal lengths and widths.
1. To find the estimated difference in length between two petals where one is 0.2 cm wider than the other, we use the regression line. Regression analysis aims to estimate the relationship between two variables, in this case, petal width and length. The correlation coefficient (r) represents the strength and direction of the linear relationship.
Since the scatter diagram is football-shaped, it indicates a curvilinear relationship, but for the purpose of this question, we'll assume the correlation coefficient (r = 0.8) is a good approximation for estimating the relationship.
The regression equation to estimate the petal length (Y) based on petal width (X) is given by:
Y = a + bX
To estimate the difference in length when one petal is 0.2 cm wider than another, we substitute the width values into the regression equation and find the difference in the estimated lengths.
2. The regression estimate of the length of a petal that is 1.5 cm wide can be obtained by substituting the width value into the regression equation:
Y = a + b(1.5)
3. To determine 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 standard deviations provided. The standard deviations give us information about the spread or variability of the data.
We'll use the normal distribution to approximate the percentage.
Now, let's calculate the answers to each question:
1. To estimate the difference in length between the petals, we need to know the slope (b) of the regression line. Unfortunately, the question does not provide the slope, so we cannot calculate the exact difference in length.
2. To find the regression estimate of the length for a petal that is 1.5 cm wide, we substitute 1.5 for X in the regression equation:
Y = a + b(1.5)
3. To find the percentage of petals that are less than 2.75 cm long among those that are 1.5 cm wide, we first need to calculate the z-score for 2.75 cm using the mean and standard deviation of petal lengths:
z = (X - mean) / SD
Then, we can use the z-score to find the corresponding percentage from the standard normal distribution table.