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 will use the information provided and apply the principles of regression analysis.
1. First, we need to determine the slope of the regression line, which represents the relationship between the petal width and length. The correlation coefficient (r) is given as 0.8, indicating a strong positive relationship. The slope (b) can be obtained using the formula:
b = r * (SD(length) / SD(width))
= 0.8 * (0.25 cm / 0.2 cm)
= 1 * 1.25
= 1.25
Therefore, the wider petal is estimated to be 1.25 cm longer than the other petal.
2. To find the regression estimate of the length for a petal that is 1.5 cm wide, we will use the formula for the regression line:
length = intercept + slope * width
The intercept of the regression line can be found by substituting the average values (3.1 cm for length and 1.8 cm for width) into the equation:
intercept = average length - slope * average width
= 3.1 cm - 1.25 * 1.8 cm
= 3.1 cm - 2.25 cm
= 0.85 cm
Now, we can calculate the estimate for a 1.5 cm wide petal:
length = 0.85 cm + 1.25 * 1.5 cm
= 0.85 cm + 1.875 cm
≈ 2.72 cm
Therefore, the regression estimate for the length of a petal that is 1.5 cm wide is approximately 2.72 cm.
3. To determine the percentage of petals that are 1.5 cm wide and less than 2.75 cm long, we can use the cumulative distribution function (CDF) of the regression line. This represents the probability of observing a value less than or equal to a certain value. We can use this to estimate the percentage.
First, we need to find the z-score (standardized score) for the length value of 2.75 cm using the formula:
z = (x - average length) / SD(length)
= (2.75 cm - 3.1 cm) / 0.25 cm
= -1.4
We can then look up the corresponding area under the standard normal distribution curve for a z-score of -1.4. Using a standard normal distribution table or a calculator, we find that the area is approximately 0.0808 or 8.08%.
Therefore, about 8.08% of the petals that are 1.5 cm wide are expected to be less than 2.75 cm long.