Question

1. Here is a link to the data set collected by Dr. Hernandez. Her research team has a data set of the weight of fishes caught from both the contaminated lake and the pristine lake. Assume both of these data sets have equal variance.

   Fish Weight (g)
   Pristine	Contaminated
   45.3	79.3
   56.8	96.8
   79.6	84.6
   61.4	79.6
   39.6	97.6
   89.3	80.5
   74.9	76.9
   95.5	89.7
   71.6	86.6
   92	89.4
   76.3	72.5
   46.5	92.3
   77.6	68.9
   88.7	56.9
   70.6	99.5
   55.4	87.3
   90.3	84
   77.6	88.5
   54.6	79
   39.9	71.2
   90.5	78.8
   84.6	68.8
   68.8	61.6
   82.9	98.5
   85.2	58.4
   90.5	91.1
   86.34	97.6
   79.9	71.4
   86.7	89.6
   56	81.7

   Perform the correct statistical test and determine the P value to test Dr. Hernandez's hypothesis. Report your answer for P rounded to four decimal places.

   Use formulas, functions, and the data analysis tool pack in Excel to avoid accruing rounding errors. Useful functions include

   =t.test(array1,array2,tails,type)

   to produce a P value for any type of a t test from a data set

   Type can be set to 1 for paired, 2 for unpaired with equal variance, or 3 for unpaired with unequal variance

   =chidist(x,degrees of freedom)

   to produce a P value from a calculated chi squared statistic

   ANOVA Single Factor from the data analysis tool pack

   to perform a 1-way ANOVA on a data set with only one factor

   Regression from the data analysis tool pack

   to perform a linear regression and the corresponding t-test on the slope of the line

3 points   

QUESTION 38

1. Based on the statistic results and P value determined in the previous question, which of the following statements are FALSE? (Check all that apply)

   		The calculated statistic value would be less than the critical value (when alpha = 0.05).
   		The P value is less than 0.05.
   		The P value is greater than alpha = 0.05.
   		The calculated statistic value would be greater than the critical value (when alpha = 0.05).

1 points   

QUESTION 39

1. Which of the following statements are correct interpretations of Dr. Hernandez's research results? (Check all that apply)

   		The bovine growth hormone contamination affects the growth of fish
   		Reject the null hypothesis
   		Fish the reached maturity in the contaminated pond did not have a discernible difference in weight from fish that reached maturity in the pristine pond.
   		The bovine growth hormone contamination did not affect the growth of fish
   		Fish the reached maturity in the contaminated pond had a greater mean weight than fish that reached maturity in the pristine pond.
   		Fail to reject the null hypothesis

Answer 1

Solution:

Here, we have to use two sample t test for population means.

Null hypothesis: H₀: There is no significant difference in the average weights of the fishes caught from Pristine and Contaminated lake.

Alternative hypothesis: H_a: There is a significant difference in the average weights of the fishes caught from Pristine and Contaminated lake.

H₀: µ₁ = µ₂ versus H_a: µ₁ ≠ µ₂

This is a two tailed test.

We assume level of significance = α = 0.05

Test statistic formula for pooled variance t test is given as below:

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

From given data, we have

X1bar = 73.16467

X2bar = 81.95333

S1 = 16.8419

S2 = 11.8555

n1 = 30

n2 = 30

df = n1 + n2 – 2 = 30 + 30 – 2 = 58

Critical values = - 2.0017 and 2.0017

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(30 – 1)* 16.8419^2 + (30 – 1)* 11.8555^2]/(30 + 30 – 2)

S_p² = 212.1013

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

t = (73.16467 – 81.95333) / sqrt[212.1013*((1/30)+(1/30))]

t = -8.7887/3.7603

t = -2.3372

P-value = 0.0229

(by using t-table)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that there is a significant difference in the average weights of the fishes caught from Pristine and Contaminated lake.

QUESTION 38

Based on the statistic results and P value determined in the previous question, which of the following statements are FALSE? (Check all that apply)

The P value is greater than alpha = 0.05.

QUESTION 39

Which of the following statements are correct interpretations of Dr. Hernandez's research results? (Check all that apply)

Reject the null hypothesis

Fish the reached maturity in the contaminated pond did not have a discernible difference in weight from fish that reached maturity in the pristine pond.