Bayes Theorem

Bayes Theorem

P(B/A)=P(A and B)/P(A) but from Bayes Theorem we have
P(B/A) = P(A/B)*P(B)/P(A)

in our problem let A = identify correctly and B = cat person
therefore A' = identify incorrectly and B' = dog person

P(A) = P(A/B)*P(B) + P(A/B')P(B')

Note the tree diagram in the written work.

The values used and obtained are as follows

P(B) = .33
P(B') = .67
P(A/B) = .96
P(A'/B) = .04
P(A/B') = .71
P(A'/B') = .29
P(A and B) = .33(.76) = .3168
P(A' and B) = .33(.04) = .0132
P(A and B') = .67(.71) = .4757
P(A' and B') = .67(.29) = .1943

Notice that all the joint probabilities add to 1

Now put those values into the formula and you'll get P(B/A) = .3997

Assumptions for hypothesis test with proportions

 First we must see if np >10 and n(1-p)> 10

Also the sampling method must be a simple random sample, with only two possible outcomes, p (success) and 1-p (failure). 

The sample must include at least 10 successes and 10 failures and the population size is at least 20 times larger than the sample size.

The sample data distribution tends to resemble the population distribution more closely than the sampling distribution. A random sample of data from a population should be representative of the population, and its distribution should be similar to the population distribution.

 Suppose that we draw all possible samples of size n from a given population and then get the mean, standard deviation, proportion or other statistic for each sample. The probability distribution of this statistic is called a sample distribution.

Now suppose we take all possible samples of a certain size from a population. Once we obtain the samples, we get the mean for each sample. This is called the sampling distribution of the sample mean.

The price ceiling which is the highest the price can be is lower than the price equilibrium, so when the price is $5, you see the supply is 10 and the demand is 30. This means there is a higher demand than what is in supply, therefore there is a shortage of 20.

The price floor is the lowest the price can be and since it's still $5, the supply and demand is the same as in the first part. But since the price floor can be increased, there is no surplus or shortage, it can be moved to create equilibrium or surplus. Basically it isn't fixed, whereas with the price ceiling at $5, there is no way to rectify the situation of the shortage.

1.The distance that one professional golfer can drive a golf ball has a normal distribution with a mean of 258 yards and a standard deviation of 6 yards. What proportion of his drives exceed 280 yards in length? How many yards should this golfer drive a ball so that the distance is among his longest 25%

(X > 280) , we need to get the Z score, which is Z =(x -mean)/standard deviation

Z = (280 - 258)/6
Z = 3.67
1-Z(3.67) ,.... approximately 0, less then .0001

Z for upper 25th percent is .67

Mean + Z*standard deviation
258 +.67(6) = 262.02

2.The average number of pounds of meat a person consumes a year is 212.3 pounds. Assume that the standard deviation is 20 pounds. If a sample of 50 individuals is selected, find the probability that the mean of the sample will be less than 210 pounds per year.

 mean = 212.3, standard deviation is 20
n = 50

P(x-bar < 210)
Z = (x-bar -mean)/(standard deviation/square root(n))
Z = (210 - 212.3)/(20/square root(50))
Z = -0.81

Z(-0.81) = .2090

answer is .2090

3. Suppose that a presidential candidate is favored by 51% of all eligible voters. What is the probability that in a random sample of 100 registered voters, less than 49% will favor that candidate?

p = .51
1-p = .49

Z = (p^ - p)/(sqrt(p(1-p)/n))

Z = (.49 - .51)/(sqrt(.51*.49)/100)
Z = -0.4

Z(-0.4) = .3446