Am I on the right track with all of these

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In summary, the "right track" in science involves following the scientific method and staying current with research in your field. To know if you are on the right track, regularly seek feedback and critically evaluate your work. Signs that you may not be on the right track include conflicting results and difficulty replicating experiments. It is not uncommon for research to take unexpected turns, and staying motivated and focused can be aided by setting clear goals, seeking support, and taking breaks.
  • #1
OptimusPrime
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1.Given the joint probability distribution of Y1 and Y2:

f(y1,y2)= (2/5)*(2y1+3y2), 0
0, elsewhere

find

a) f(y1) and f(y2) the marginal distributions

b) Given: E(y1y2) = 1/3 E(y1)=17/30, E(y2)=3/5

then Cov(y1,y2)=?

c) If E(y1^2) = 7/18 and E(y2^2) = 4/9 and E(y1)=17/30 and E(y2)=3/5
Find the correlation coeficcient. Comment on the strength of the correlation coefficient.

r(y1,y2)= Cov(y1,y2) / [Sd(y1) * SD(y2)]


So am I on the right track. Can anyone help me please?
a)
f(y1)=int(f(y1,y2),y2,0,1)
f(y2)=int(f(y1,y2),y1,0,1)

b)
Cov(Y1,Y2)=E(Y1Y2)-E(Y1)E(Y2)
Cov(Y1,Y2)=1/3-(17/30)(3/5)

c)
need SD(Y1) and SD(Y2)

V(Y1)=E(Y1^2)-[E(Y1)]^2
V(Y2)=E(Y2^2)-[E(Y2)]^2

so the correlation is weak.


2.Scores on an exam are assumed to be normally distrubuted with a mean of 78 and variance of 36

a) What is the probability that a person taking the exam scores higher than 75?

b) Suppose the student socring in the top 10% of this distribution are to receive an A grade, what is the minimum score that a student must achieve to earn an A grade?

c) What must be the cut off point for passing the exam if the examiner wants only 30% of all scores to be passing?

d) Approximately, what proportion of the students have scores 5 or more points above the score that cuts off the lowest 25 %?

This is what I did.

All: Mean = 78
All: Variance = 36, then Standard Deviation = 6

a. (75-78)/6 = -1/2 -- That is one-half standard deviation below the mean. I get 80.85%

b. find a score, S, such that (S - 78)/6 = 1.2815516.
S= 1.28*6+78
c)have to Find the 70th percentile of the standard normal distribution and translate to a grade.
d) have to find the first quartile (25th percentile) of the standard normal distribution, translate to a grade, and add 5 points

Any help would be appreciated
tHANKS!
 
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  • #2
OptimusPrime said:
1.Given the joint probability distribution of Y1 and Y2:

f(y1,y2)= (2/5)*(2y1+3y2), 0
0, elsewhere

find

a) f(y1) and f(y2) the marginal distributions

b) Given: E(y1y2) = 1/3 E(y1)=17/30, E(y2)=3/5

then Cov(y1,y2)=?

c) If E(y1^2) = 7/18 and E(y2^2) = 4/9 and E(y1)=17/30 and E(y2)=3/5
Find the correlation coeficcient. Comment on the strength of the correlation coefficient.

r(y1,y2)= Cov(y1,y2) / [Sd(y1) * SD(y2)]


So am I on the right track. Can anyone help me please?
a)
f(y1)=int(f(y1,y2),y2,0,1)
f(y2)=int(f(y1,y2),y1,0,1)

Yes, that's the right definition (assuming you meant that y1 and y2 are between 0 and 1. What did you get when you actually integrated?

b)
Cov(Y1,Y2)=E(Y1Y2)-E(Y1)E(Y2)
Cov(Y1,Y2)=1/3-(17/30)(3/5)

I haven't checked your calculation but that looks good.

c)
need SD(Y1) and SD(Y2)

V(Y1)=E(Y1^2)-[E(Y1)]^2
V(Y2)=E(Y2^2)-[E(Y2)]^2

so the correlation is weak.

OK

2.Scores on an exam are assumed to be normally distrubuted with a mean of 78 and variance of 36

a) What is the probability that a person taking the exam scores higher than 75?

b) Suppose the student socring in the top 10% of this distribution are to receive an A grade, what is the minimum score that a student must achieve to earn an A grade?

c) What must be the cut off point for passing the exam if the examiner wants only 30% of all scores to be passing?

d) Approximately, what proportion of the students have scores 5 or more points above the score that cuts off the lowest 25 %?

This is what I did.

All: Mean = 78
All: Variance = 36, then Standard Deviation = 6

a. (75-78)/6 = -1/2 -- That is one-half standard deviation below the mean. I get 80.85%

When I check a table of "Normal Distribution" values (http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/normaltable.html [Broken] is a good one), I get that the area under the curve, from 0 to 0.5 is 0.1915 and so this should be 69.15%, not 80.85%. You seem to have subtracted .1915 from 1 which would be the percentage of scores above 78+3= 81 or below 78.

b. find a score, S, such that (S - 78)/6 = 1.2815516.
S= 1.28*6+78

yes, that's what I get.

c)have to Find the 70th percentile of the standard normal distribution and translate to a grade.

Yes. Again, 70% corresponds to z= 0.52 approx. What grade, S, is that?

d) have to find the first quartile (25th percentile) of the standard normal distribution, translate to a grade, and add 5 points

The table tells me that the area under the normal curve from 0 to .675 (approx) is 0.25. By symmetry, the area from -infinity to -.675 is 0.25 so the first quartile corresponds to z= -.25. S= 78+ 6(-.25)= 76.5. 5 points above that is 81.5.

Any help would be appreciated
tHANKS!

Having typed all this in, I got an error message saying "The message you have entered is too short. Please lengthen your message to at least 10 characters."! Well, this is more than 10 characters right here, isn't it?
 
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  • #3



Yes, you are on the right track with your calculations. Here are some tips to help you check your work and make sure you are on the right track:

a) For finding the marginal distributions, you are correct in integrating over the other variable and setting its limits to 0 and 1. However, make sure you also divide by the total probability (in this case, it is 1) to get the correct marginal distribution.

b) Your calculation for the covariance is correct. Just make sure to use the correct values for E(y1) and E(y2) that are given in the problem.

c) To find the correlation coefficient, you will also need to find the standard deviations of Y1 and Y2. You can do this by taking the square root of the variances that are given in the problem. Then, use the formula r(y1,y2)= Cov(y1,y2) / [Sd(y1) * SD(y2)] to find the correlation coefficient.

For the second problem:

a) To find the probability that a person scores higher than 75, you will need to use the standard normal distribution. First, convert the score of 75 to a z-score using the formula z = (x - mean) / standard deviation. Then, use a z-table or a calculator to find the probability that a z-score is greater than the z-score you calculated.

b) To find the minimum score for an A, you will need to find the z-score that corresponds to the top 10% of the standard normal distribution. Then, use the z-score formula to find the score that corresponds to that z-score.

c) To find the cut off point for passing, you will need to find the z-score that corresponds to the 30th percentile of the standard normal distribution. Then, use the z-score formula to find the score that corresponds to that z-score.

d) To find the proportion of students with scores 5 or more points above the score that cuts off the lowest 25%, you will need to find the z-score that corresponds to the 25th percentile of the standard normal distribution. Then, add 5 points to that score and find the z-score that corresponds to the new score. Finally, use the z-score formula to find the proportion of students with scores higher than that z-score.

Overall, it looks like you are on the right track with both problems. Just make sure to double check your
 

1. What is the "right track" in the context of science?

The "right track" in science refers to following the scientific method, which involves asking questions, formulating hypotheses, conducting experiments, and analyzing data to draw conclusions. It also involves staying up-to-date with current research and peer-reviewed studies in your field.

2. How can I know if I am on the right track with my research?

You can know if you are on the right track with your research by regularly checking in with your mentor or colleagues, presenting your findings at conferences or meetings, and receiving feedback from experts in your field. It is also important to critically evaluate your own work and ensure that it aligns with the current scientific knowledge.

3. What are some signs that I am not on the right track with my research?

Some signs that you may not be on the right track with your research include conflicting or inconclusive results, lack of support from other scientists in your field, and difficulty replicating your experiments. It is important to critically evaluate any potential flaws or biases in your methods and results.

4. Should I be concerned if my research takes me in a different direction than I initially planned?

No, it is not uncommon for scientific research to take unexpected turns and for initial hypotheses to be disproven. In fact, unexpected results can often lead to new discoveries and advancements in the field. As long as you are following the scientific method and critically evaluating your findings, you are on the right track.

5. How can I stay motivated and focused when conducting research?

Staying motivated and focused during research can be challenging, but there are some strategies that can help. These include setting clear goals and deadlines, breaking down large tasks into smaller, manageable ones, seeking feedback and support from colleagues, and taking breaks to recharge and prevent burnout. It is also important to remember the potential impact and importance of your research in contributing to the greater scientific community.

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