Question

Use the following regression output from R studio and answer the following questions.

**** **** myresults1 <- Im(testlDat$Test3 testlDat$Test1) > summary (TestDat. Im Im formula = test1Dat$Test3 ~ testlDat$Testi) Coefficients: Estimate Std. Error t value pr(>It!) (Intercept) 8.05912 1.90816 4.224 0.000107 testlDat$Testi 0.78128 0.05445 (Residual standard error: 4.228 on 48 degrees of freedom Multiple R-squared: 0.8109, Adjusted R-squared: 0.807 F-statistic: 205.9 on 1 and 48 Dep-value: < 2.2e-16 > confint(myresults1) 2.5 % 97.5 % (Intercept) 4.2225059 11.8957280 testlDat $Test1 0.6718002 0.8907529 Scatterplot showing relation between Test 1 and Test 3 Marks Histogram of Residuals 45 SC Test3 Frequency 10 15 20 05 25 15 -10 10 15 10 20 30 40 50 TestDatres Test

1. Comment on the relation between the two test marks.

A. There appears to be no  linear relation between the two test marks
B. There appears to be strong negative linear relation between the two test marks.
C. There appears to be weak positive linear relation between the two test marks.
D. There appears to be strong positive  linear relation between the two test marks.
2. If the linear equation between the test 3 and test 1marks is given as:
What is the value of a from the output above Answer
What is the value of b from the output above Answer
Is there a relation between Test3 and Test1 Marks for SIBT 2016 Math102 students?
Perform an appropriate hypothesis test to answer the above research question:
2. The null and alternative hypotheses are
A. H0: μ = 0, H1: μ ≠ 0
B. H0: β = 0, H1: β ≠ 0
C. H0: μ = 0, H1: μ ≠ 0
D. H0: π = 0, H1: π ≠ 0
E. H0: μ1- μ2 = 0, H1: μ1- μ2 ≠ 0
.
3.   How can you justify the assumption for this test?
A. Since nπ and n(1-π) are both ≥5, the Central Limit Theorem applies and we can assume that p arises from a normal distribution
B. The shape of the histogram indicates that the scores may be from a normal distribution and/or the sample size is reasonably large so the sample mean will be from a normal population
C. The boxplots indicate that the samples are fairly symmetric and the spreads are similar, so we can assume both sets of data come from normal distributions with the same spread
D. The points are randomly scattered around a straight line, so the assumptions of linearity and constant spread are satisfied. The shape of the histogram indicates that the residuals come from a normal distribution
E. None of these

4. What is the value of the t test statistic?:
A. 4.224 B. 205.9 C. 14.35 D. 0.78128 E. 0.05445

5. What are the degrees of freedom?
A. 46 B. 26 C. 52 D. 50 E. 48
  
6. What is the correct decision for this test?
A. Since p-value < 0.05, reject H0
B. Since p-value > 0.05, reject H0
C. Since p-value < 0.05, do not reject H0
D. Since p-value > 0.05, do not reject H0
7. An appropriate conclusion for this test is:
A. There is a significant positive linear relation between Test3 and Test1 marks and If Test1 mark increases by 1 point,  we expect Test 3 mark to increase by 0.78 points, on average.
B. There is a significant positive linear relation between Test3 and Test1 marks and If Test3 mark increases by 1 point,  we expect Test1 mark to increase by 8.05912 points, on average.
C. There is a significant positive linear relation between Test3 and Test1 marks and If Test3 mark increases by 1 point,  we expect Test 1 mark to increase by 0.78 points, on average.
D. There is a significant negative linear relation between Test3 and Test1 marks and If Test1 mark increases by 1 point,  we expect Test 3 mark to increase by 0.78 points, on average.
E. There is a significant positive linear relation between Test3 and Test1 marks and If Test1 mark increases by 1 point,  we expect Test 3 mark to increase by 8.05912 points, on average.
8. What is the value of the correlation coefficient? (2 decimal places)
Answer
9. How would you interpret the correlation coefficient.
A. There is a weak positive linear relation between test1 and Test3
B. There is a very strong negative non-linear relation between test1 and Test3
C. There is a very strong positive non-linear relation between test1 and Test3
D. There is a very strong negative linear relation between test1 and Test3
E. There is a very strong positive linear relation between test1 and Test3
10. Can we use the regression to predict test 3 marks when test 1 mark is 60?
A. Yes, we can make prediction as the value 60 is within the range of the data
B. No, we can only predict test 1 mark from test 3 no the other way round
C. No, we can't make prediction as there is no linear relation exist
D. No, because we did not reject the null hypothesis
E. No, we can't make prediction as the value 60 is out of the range of the data
11. Predict the Test 3 marks if Test 1 mark is 40. ( 2 decimal places)
Answer

Answer 1

Que.1

There appears to be strong positive  linear relation between the two test marks.

Because if we increase marks of test 1 then marks of test 3 also increases.

2.

Estimated regression equation is,

Test 3 = 8.0591 + 0.7813 test 1

The value of b from the output above Answer is 0.7813

Hypothesis:

H0: β = 0, H1: β ≠ 0

3.

The points are randomly scattered around a straight line, so the assumptions of linearity and constant spread are satisfied. The shape of the histogram indicates that the residuals come from a normal distribution

4.

The value of the t test statistic = 0.78128 / 0.05445 = 14.35

5.

Degrees of freedom = 48 (error degrees of freedom given for F test)

6.

P-value = 2.2e-16

It is obtained using excel with function =TDIST(14.35,48,2)

Since p-value < 0.05, reject H0

7.

There is a significant positive linear relation between Test3 and Test1 marks and If Test1 mark increases by 1 point,  we expect Test 3 mark to increase by 0.78 points, on average.

8.

Correlation coefficient:

$\small r=\sqrt{R^2} = \sqrt{0.8109} = 0.90$

9.

There is a very strong positive linear relation between test1 and Test3

10.

Yes, we can make prediction as the value 60 is within the range of the data

11.

Test 3 = 8.0591 + 0.7813 test 1

Test 3 = 8.0591 + 0.7813 (40)

Test 3 = 39.31