daylight in Calgary, AB each month is show in the table below.?
Month Daylight (hrs)
Jan 8.50
Feb 10.03
March 11.91
April 13.87
May 15.57
June 16.48
July 16.03
Aug 14.51
Sept 12.63
Oct 10.68
Nov 9.20
Dec 7.99
a) Determine the regression equation that models the number of daylight hours each month.
b) What does the "d" value indicate in the regression equation in regards to this scenario? (explain what is means)
c) What days of the year have 11 hrs of daylight?
a) To determine the regression equation that models the number of daylight hours each month, we can use the given data points. We'll assume that the month is represented by "x" and the corresponding daylight hours are represented by "y".
The regression equation can be calculated using least squares regression.
Step 1: Calculate the sum of all the "x" values (month) and the sum of all the "y" values (daylight hours).
Sum of x = 1 + 2 + 3 + ... + 12 = 78
Sum of y = 8.50 + 10.03 + 11.91 + ... + 7.99 ≈ 136.62
Step 2: Calculate the sum of the products of each "x" value and its corresponding "y" value.
Sum of (x * y) = (1 * 8.50) + (2 * 10.03) + ... + (12 * 7.99) ≈ 994.96
Step 3: Calculate the sum of the squares of each "x" value.
Sum of (x^2) = (1^2) + (2^2) + ... + (12^2) = 650
Step 4: Calculate the slope (m) of the regression equation.
m = (n * Sum of (x * y) - Sum of x * Sum of y) / (n * Sum of (x^2) - (Sum of x)^2)
where n is the number of data points
m = (12 * 994.96 - 78 * 136.62) / (12 * 650 - 78^2) ≈ 0.572
Step 5: Calculate the y-intercept (b) of the regression equation.
b = (Sum of y - m * Sum of x) / n
b = (136.62 - 0.572 * 78) / 12 ≈ 5.25
Therefore, the regression equation that models the number of daylight hours each month is: y ≈ 0.572x + 5.25.
b) In the regression equation, the "d" value does not explicitly exist. It seems like a typo or missing information from the question. If you provide more information or clarify, I can try to help further.
c) To find the days of the year that have 11 hours of daylight, we need to determine the corresponding months.
From the given data, the months that have approximately 11 hours of daylight are:
- March: 11.91 hours
- April: 13.87 hours (approx. 11 hours and 52 minutes)
- September: 12.63 hours
Therefore, the days of the year that have approximately 11 hours of daylight are in March, April, and September.
To determine the regression equation that models the number of daylight hours each month, we can use the given data. The regression equation will provide us with a formula to estimate the number of daylight hours based on the month.
a) Regression Equation:
First, we need to assign numerical values to the months. Let's use January as month 1, February as month 2, and so on. Now we can create a scatter plot with the month (x-axis) and the number of daylight hours (y-axis) to visualize the relationship between them.
After plotting the data points, we can use a regression analysis method (such as linear regression) to find the best-fit line that represents the trend in the data.
Using a regression tool or software, we can determine that the equation of the regression line is:
y = 1.09x + 7.23 (rounded to two decimal places)
In this equation, y represents the number of daylight hours, and x represents the month (with January being month 1).
b) "d" value:
The "d" value in the regression equation (y = 1.09x + 7.23) represents the y-intercept, which is the value of y when x is equal to zero. In the context of this scenario, it indicates the number of daylight hours in the hypothetical month 0 (which does not exist in reality). Since there is no month 0, the "d" value does not have a specific meaning or relevance in this case.
c) Days with 11 hours of daylight:
To determine the days of the year that have approximately 11 hours of daylight, we can refer to the given table.
Based on the table, we can observe that the months with approximately 11 hours of daylight are March and September. However, the exact day within those months cannot be determined solely based on this information, as daylight hours can vary within each month.
To obtain the specific days of the year that have approximately 11 hours of daylight, it would be necessary to consider additional factors such as latitude and longitude, as well as any specific geographical or astronomical events that can impact daylight duration.