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Did you read my tips on how to study and learn this course? If not, here is a link to those important suggestions: The department posted solutions for Assignment 1 (these are not my solutions). Click here.The department posted solutions for Assignment 2 (these are
not my solutions). Click here.Here is a link to the actual assignment, in case you don't have it handy: Study Lesson 5: Introduction to Probability, Lesson 6: The Binomial Distribution and Lesson 7: The Distribution of the Sample Mean in my Basic Stats 1 book to prepare for this
assignment. NOTE: You need only study to the end of question 7 in Lesson 7 (end of page 447). The section on control charts has been removed from the course.
Of course, always seek out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the
test.
Although this question appears to be a 3-circle Venn diagram problem (similar to my Lesson 6, question 18), they have left key information out that renders it not worth the effort to try to make the diagram right away.
To make a three-circle Venn diagram, for event A, B and C, among the givens you need the following:
- The probability an outcome belongs to all three of A, B and C.
- The three "and" pairs, which is to say, the P(A and B), the P(A and C), and the P(B and C).
- Then you can assume that everything will go smoothly filling in the 8 sections of the
Venn diagram.
If the above is not given, you should use the probability formulas to solve what is missing above first.
In this problem, you have been given the probability an outcome belongs to all three of J, B, and G; you have been given P(J and B); and, you have been given P(J and G). The problem is that you have not been given P(B and G). Before I make a Venn, I would use a probability formula to solve that missing value.
But that is actually one of their questions below, so, you are better off just taking each question as it comes and answering it directly.
Part (a) This is tailor-made for the General Addition Rule P(A or B) = P(A) + P(B) - P(A and B). You have been given everything you need.
Part (b)
Again, use the General
Addition Rule P(A or B) = P(A) + P(B) - P(A and B) and a little algebra. Note that you have been given P(J or G), P(J and G), and P(J). So, use those to figure out P(G).
Part (c)
Now, they want P(B and G).
Don't you dare use the formula P(B and G) = P(B)*P(G)!!!!
THAT FORMULA IS ONLY TRUE
WHEN TWO EVENTS ARE INDEPENDENT!!!
You have no idea whether B and G are independent or not.
One of the most common mistakes students make, is thinking that, anytime they want to know P(A and B), they merely have to compute P(A)*P(B). WRONG!!!! You can only do that when you are told that A and B are INDEPENDENT.
Here is the proper thing to
do: The General Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), is ALWAYS TRUE. Use that formula to solve either "or" or "and" unknowns whenever possible. That formula has four terms: P(A), P(B), P(A and B), and P(A or B). If you are given any three of those four terms, you can use this formula and a little algebra to solve the missing fourth term. That is what you should have done to solve part (b) above, and that is what can also be
done here to solve part (c). You can solve P(B and G) if you know P(B), P(G) and P(B or G).
Part (d)
Now, you should know P(G) from part (b) and P(B and G) from part (c). Now you can prove whether any two events in this problem are independent.
Two events are independent if and only if they satisfy the Independence Multiplication Rule,
P(A and B) = P(A)*P(B).
For example, you were told P(J)=0.55, P(B)=0.40, and P(J and B)=0.27.
If J and B are independent, then P(J and B) = P(J)*P(B).
0.27 should equal 0.55*0.40.
But 0.55*0.40 = 0.22, not 0.27, proving that J and B are dependent, not independent.
You can do a similar test for the other pairs.
Part (a)
I would use a two-way table to set up and solve this problem. Similar to what I do in Lesson 5. Note that they want the probability distribution of X, the number of heads, similar to my questions 3(f) and
4(f) where I use the earlier two-way table and distribution I had come up with to then do a probability distribution for X, the number of boys.
Part (b) This certainly can be solved using binomial methods. Keep in mind that X is the number of heads. What is n and p? They then want you to solve the probabilities for X=0, X=1, etc. using the binomial formula. Obviously, those probabilities should match your answers in part
(a).
This is Lesson 7 stuff. You have to always be asking yourself:
- Is the problem talking about one individual score X? Or, is it talking about the mean of n scores,
x-bar?
- If it is talking about just one score X, is X normally distributed? Only if X is normally distributed can you compute the probabilities of some region for X, since only then can we standardize X it a z score.
- If it is talking about the mean of n scores, x-bar, can we assume x-bar is normally distributed? Why or why not? Remember, if a population is known to be
normal, then x-bar has a normal distribution for any sample size, n. If a population is not normal, it is still possible that the distribution of the sample mean, x-bar, is normal, but n must be large.
- Central Limit Theorem: If n is large, the distribution of x-bar, the sample mean, is approximately normal.
- If x-bar is normal (even approximately) then we can standardize x-bar into a z-score.
If you are told for a
fact that a distribution is normal, then you can compute exact probabilities because you know the exact values for the normal distribution. And if you know that X is normal, then it also follows that x-bar is normal, too. So, you can compute exact probabilities for X or x-bar, when the given population is normal.
If a population is not normal, then, you may be able to assume x-bar is approximately normal by Central Limit
Theorem. However, if x-bar is only approximately normal, you will only be able to compute approximate probabilities.
Look at my questions 4 and 5 in Lesson 7 for examples.
IMPORTANT CALCULATOR TIPS: When computing things like sigma/ square root n, make sure you round off to NO LESS THAN 4 decimal
places. Better yet, store the exact value in the memory of your calculator.
Look at my questions 6 and 7 in Lesson 7 for examples. Note that they give you a total amount in the question.
IMPORTANT CALCULATOR TIPS: When
computing things like sigma/ square root n, make sure you round off to NO LESS THAN 4 decimal places. Better yet, store the exact value in the memory of your calculator.
More x-bar bell curve stuff like the previous two questions. Note that question 11 requires that you work backwards since you are given the probability and can read the table backwards to get a z score and solve the unknown algebraically.
Be careful, they are talking about the mean of a sample of some size n=4. Thus, you must compute the mean of x-bar and the standard deviation of x-bar, which are mu and sigma
divide by square root n first, then use those values for your 68-95-99.7 rule. Make sure you put the smaller answer first.
This is p^ bell curve stuff similar to what I do in Lesson 6, question 10(c) and (d).
Be careful to note which is the true proportion, p, and which is the sample proportion p^.
Be careful that you
don't lose accuracy by rounding off too much. I suggest you round off to no less than 5 or 6 decimal places while computing things like the standard deviation of p^ to ensure that you get accurate z-scores. Better yet, store exact answers in memory in your calculator.
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