Question

2a) Calculate an average and sample standard deviation for the independent variable (time of heating) and the dependent variable (number of surviving organisms) given the data in the table below.

Time (min) ... Number of Survivors

0.1 ...    2.01x10^6

7.5 ... 2.95x10^5

15 ...    8.42x10^4

22.5 ... 2.43x10^4

30 ... 6.99x10^3

2 b) Plot the number of survivors over tie on Cartesian, semi-log and log-log axes using only the data points. Identify the plot that best presents the data. Explain why this plot is better. Add a trend line that presents the data, including an R^2 value and equation.

2 c) Using the prediction equation developed in 2b, estimate the number of survivors after heating for 50 minutes. How confident is this estimate? Why?

Answer 1

2A and 2B

First I shall plot the graphs in cartesian co-ordinates

Now in the semi-logarithmic scale

Now I shall use logarithm for both the time and the number of survivors

From the above three plots, we can conclude that the log of the number of survivors vs time will be the best plot.

The trend line is fitted below,

The R squared value will be calculated from the excel regression equation

the R-squared value is calculated as follows:

Regression Statistics
Multiple R	0.995004438
R Square	0.990033831
Adjusted R Square	0.986711775
Standard Error	0.109909305
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	1	3.600078535	3.600078535	298.0183808	0.000423531
Residual	3	0.036240166	0.012080055
Total	4	3.636318701
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	6.190504392	0.085362964	72.51979185	5.77835E-06	5.918841342	6.462167442	5.918841342	6.462167442
X Variable 1	-0.080214492	0.004646557	-17.26320888	0.000423531	-0.095001912	-0.065427072	-0.095001912	-0.065427072

From the above results, we can see that the R-squared value will be 0.99 and the adjusted R-squared value will be 0.9867.

2C. If we use the regression equation, then the estimated number of survivors will be:

2.179779795

The confidence we have chosen here is 95%. That means, we are 95% sure that the number of survivors will be the same as stated.

Reason: As we have modelled the regression equation and p-values are less than 0.05, hence we can conclude that the coefficients are significant and the coefficients can be used to predict the number of the survivor with 95% confidence. Hence the statement is validated.

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