Answer:
(a)
(b)
Step-by-step explanation:
Given
Solving (a): The pmf
This means that we list out the probability of each value of x.
To do this, we simply subtract the current probability value from the next.
So, we have:
The calculation is as follows:
The x values are gotten by considering where the equality sign is in each range.
means
means
means
means
means
Solving (b):
This is calculated as:
From the given function
So:
This is calculated as:
Answer:
The Depth of water in the bathtub initially = 2 cm
Step-by-step explanation:
For this question, we want to f8nd the amount of water present in the bathtub at time t = 0 using line of best fit, that is, linear regression analysis.
The table provided shows that the depth of water in the bathtub (y) changes as time (x) progresses.
Time (min) (x) | 2 | 4 | 6 | 8 | 10 | 12 | | 14
Depth (cm) (y) | 8 | 15 | 29 | 37 | 39 | 49 | 55
Running the analysis on a spreadsheet application, like excel, the table of parameters is obtained and presented in the first attached image to this solution.
Σxᵢ = sum of all the independent variables (sum of all the time data)
Σyᵢ = sum of all the dependent variables (sum of all the depth data)
Σxᵢyᵢ = sum of the product of each dependent variable and its corresponding independent variable
Σxᵢ² = sum of the square of each independent variable (time data)
Σyᵢ² = sum of the square of each dependent variable (depth data)
n = number of variables = 7
The scatter plot and the line of best fit is presented in the second attached image to this solution
Then the regression analysis is then done
Slope; m = [n×Σxᵢyᵢ - (Σxᵢ)×(Σyᵢ)] / [nΣxᵢ² - (∑xi)²]
Intercept b = [Σyᵢ - m×(Σxᵢ)] / n
Mean of x = (Σxᵢ)/n
Mean of y = (Σyᵢ) / n
Sample correlation coefficient r: r =
[n*Σxᵢyᵢ - (Σxᵢ)(Σyᵢ)] ÷ {√([n*Σxᵢ² - (Σxᵢ)²][n*Σyᵢ² - (Σyᵢ)²])}
And -1 ≤ r ≤ +1
All of these formulas are properly presented in the third attached image to this answer
The table of results; mean of x, mean of y, intercept, slope, regression equation and sample coefficient is presented in the fourth attached image to this answer.
The linear regression equation obtained is then
y = 3.911x + 1.857
at a regression coefficient of 0.987 (the closer the value is to 1, the more accurate the equation obtained)
So, the Depth of water in the bathtub initially, that is, when t = x= 0,
y = 3.911x + 1.857
y = 3.911(0) + 1.857 = 1.857 cm = 2 cm to the nearest integer.
Hope this Helps!!!
