How do I change this into the simplest form?
1 answer:
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It might help to draw out a probability tree. Check out the diagram below. The first two branches at the top represent bags A and B. The probability of picking either bag is 1/2 assuming each bag is equally likely to be picked.
If we go with bag A, then the probability of getting blue is 3/11 since there are 3 blue out of 8+3 = 11 total. The probability of getting bag A and a blue marble is (1/2)*(3/11) = 3/22 which I've marked in the diagram as well.
If we go with bag B, then the probability of getting blue is 7/12 since there are 7 blue out of 5+7 = 12 total. The probability of getting bag B and blue is (1/2)*(7/12) = 7/24
Add up the results found getting
P(blue) = P(blue & bag A) + P(blue & bag B)
P(blue) = 3/22 + 7/24
P(blue) = 36/264 + 77/264
P(blue) = 113/264
This is shown in the diagram as well.
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Recall that conditional probability in general is defined as follows
P(A given B) = P(A and B)/P(B)
In this problem, we'll have
P(bag B given blue) = P(bag B and blue)/P(blue)
Therefore, we can say...
P(bag B given blue) = P(bag B and blue)/P(blue)
P(bag B given blue) = (7/24)/(113/264)
P(bag B given blue) = (7/24)*(264/113)
P(bag B given blue) = (7*264)/(24*113)
P(bag B given blue) = (7*24*11)/(24*113)
P(bag B given blue) = (7*11)/(113)
P(bag B given blue) = 77/113
In short, the probability of getting bag B, given the marble is blue, is 77/113.
Answer:
a)
If we compare the p value and the significance level given we see that
so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the we don't have a significant effect for the new treatment at 5% of significance.
b)
If we compare the p value and the significance level given we see that
so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the we have a significant effect for the new treatment at 5% of significance.
c) When we increase the sample size we increse the probability of rejection of the null hypothesis since the z score tend to increase when the sample size increase.
Step-by-step explanation:
Data given and notation
Part a: If the sample consists of n=16 individuals, are the data sufficient to conclude that the treatment has a significant effect using a two-tailed test with alpha = 0.05.
represent the sample mean
represent the sample standard deviation
sample size
represent the value that we want to test
represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the mean is different from 30, the system of hypothesis are :
Null hypothesis:
Alternative hypothesis:
Since we don't know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:
(1)
t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
P-value
First w eneed to find the degrees of freedom given by:
Since is a two-sided test the p value would given by:
Conclusion
If we compare the p value and the significance level given we see that
so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the we don't have a significant effect for the new treatment at 5% of significance.
Part b: If the sample consists of n=36 individuals, are the data sufficient to conclude that the treatment has a significant effect using a two-tailed test with alpha = 0.05.
Calculate the statistic
We can replace in formula (1) the info given like this:
P-value
First w eneed to find the degrees of freedom given by:
Since is a two-sided test the p value would given by:
Conclusion
If we compare the p value and the significance level given we see that
so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the we have a significant effect for the new treatment at 5% of significance.
Part c; Comparing your answer for parts a and b, how does the size of the sample influence the outcome of a hypothesis test
When we increase the sample size we increse the probability of rejection of the null hypothesis since the z score tend to increase when the sample size increase.