
Concept explainers
a.
Find the
a.

Answer to Problem 53CR
A one day increase in elapsed time will decrease the biomass concentration by
Explanation of Solution
Calculation:
It is given that y is the green biomass concentration
The simple linear equation is given as follows:
The above regression equation can be interpreted as, 1 unit increase of x will lead to decrease of 0.64 units in y, due to the negative slope of x.
Thus, 1 day increase in elapsed time will decrease the biomass concentration by
b.
Find the predicted value of biomass concentration if elapsed time is 40 days.
b.

Answer to Problem 53CR
The predicted value of biomass concentration for an elapsed time of 40 days is
Explanation of Solution
Calculation:
From Part (a), the regression equation is given by:
Substitute
Thus, the predicted value of biomass concentration for elapsed time of 40 days is
c.
Check whether there is a linear relationship between the two variables or not.
c.

Answer to Problem 53CR
There is convincing evidence of a useful linear relation between elapsed time and biomass concentration.
Explanation of Solution
Calculation:
It is given that the coefficient of determination
Denote
The null and alternative hypotheses are given by:
That is, there is no linear relationship between x and y.
That is, there is a linear relationship between x and y.
Assume that the level of significance for the test is
The test statistic for t–test when coefficient of determination is known is given by:
The quantity,
Here, the r value can be considered as negative, because the regression coefficient has negative value. Thus, the acceptable value of r is,
By substituting the values in the test statistic formula, one can get the value of test statistic.
That is,
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot.
- Choose View Probability > OK.
- From Distribution, choose ‘t’ distribution.
- Enter the Degrees of freedom as 56.
- Click the Shaded Area tab.
- Choose X Value and Both Tails for the region of the curve to shade.
- Enter the X value as –7.047.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the above graph, P-value is given by 1.4471E-09, which is approximately 0.
Decision rule:
Reject
Conclusion:
Here, the P-value is 0.
Therefore, the P-value is less than 0.05.
Hence reject
Thus, there is convincing evidence of a useful linear relation between elapsed time and biomass concentration.
Want to see more full solutions like this?
Chapter 13 Solutions
Introduction to Statistics and Data Analysis
- For a binary asymmetric channel with Py|X(0|1) = 0.1 and Py|X(1|0) = 0.2; PX(0) = 0.4 isthe probability of a bit of “0” being transmitted. X is the transmitted digit, and Y is the received digit.a. Find the values of Py(0) and Py(1).b. What is the probability that only 0s will be received for a sequence of 10 digits transmitted?c. What is the probability that 8 1s and 2 0s will be received for the same sequence of 10 digits?d. What is the probability that at least 5 0s will be received for the same sequence of 10 digits?arrow_forwardV2 360 Step down + I₁ = I2 10KVA 120V 10KVA 1₂ = 360-120 or 2nd Ratio's V₂ m 120 Ratio= 360 √2 H I2 I, + I2 120arrow_forwardQ2. [20 points] An amplitude X of a Gaussian signal x(t) has a mean value of 2 and an RMS value of √(10), i.e. square root of 10. Determine the PDF of x(t).arrow_forward
- In a network with 12 links, one of the links has failed. The failed link is randomlylocated. An electrical engineer tests the links one by one until the failed link is found.a. What is the probability that the engineer will find the failed link in the first test?b. What is the probability that the engineer will find the failed link in five tests?Note: You should assume that for Part b, the five tests are done consecutively.arrow_forwardProblem 3. Pricing a multi-stock option the Margrabe formula The purpose of this problem is to price a swap option in a 2-stock model, similarly as what we did in the example in the lectures. We consider a two-dimensional Brownian motion given by W₁ = (W(¹), W(2)) on a probability space (Q, F,P). Two stock prices are modeled by the following equations: dX = dY₁ = X₁ (rdt+ rdt+0₁dW!) (²)), Y₁ (rdt+dW+0zdW!"), with Xo xo and Yo =yo. This corresponds to the multi-stock model studied in class, but with notation (X+, Y₁) instead of (S(1), S(2)). Given the model above, the measure P is already the risk-neutral measure (Both stocks have rate of return r). We write σ = 0₁+0%. We consider a swap option, which gives you the right, at time T, to exchange one share of X for one share of Y. That is, the option has payoff F=(Yr-XT). (a) We first assume that r = 0 (for questions (a)-(f)). Write an explicit expression for the process Xt. Reminder before proceeding to question (b): Girsanov's theorem…arrow_forwardProblem 1. Multi-stock model We consider a 2-stock model similar to the one studied in class. Namely, we consider = S(1) S(2) = S(¹) exp (σ1B(1) + (M1 - 0/1 ) S(²) exp (02B(2) + (H₂- M2 where (B(¹) ) +20 and (B(2) ) +≥o are two Brownian motions, with t≥0 Cov (B(¹), B(2)) = p min{t, s}. " The purpose of this problem is to prove that there indeed exists a 2-dimensional Brownian motion (W+)+20 (W(1), W(2))+20 such that = S(1) S(2) = = S(¹) exp (011W(¹) + (μ₁ - 01/1) t) 롱) S(²) exp (021W (1) + 022W(2) + (112 - 03/01/12) t). where σ11, 21, 22 are constants to be determined (as functions of σ1, σ2, p). Hint: The constants will follow the formulas developed in the lectures. (a) To show existence of (Ŵ+), first write the expression for both W. (¹) and W (2) functions of (B(1), B(²)). as (b) Using the formulas obtained in (a), show that the process (WA) is actually a 2- dimensional standard Brownian motion (i.e. show that each component is normal, with mean 0, variance t, and that their…arrow_forward
- The scores of 8 students on the midterm exam and final exam were as follows. Student Midterm Final Anderson 98 89 Bailey 88 74 Cruz 87 97 DeSana 85 79 Erickson 85 94 Francis 83 71 Gray 74 98 Harris 70 91 Find the value of the (Spearman's) rank correlation coefficient test statistic that would be used to test the claim of no correlation between midterm score and final exam score. Round your answer to 3 places after the decimal point, if necessary. Test statistic: rs =arrow_forwardBusiness discussarrow_forwardBusiness discussarrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning


