Q.4 a) If a= x+y and b= x-y , show that I) (a+b)²= 4x² ii) ( a-b)²= 4y²
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Answer:
Assuming:
H0:
H1:
Step-by-step explanation:
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Let X the random variable of interest (number of correct answers in the test), on this case we now that:
The probability mass function for the Binomial distribution is given as:
Where (nCx) means combinatory and it's given by this formula:
We need to check the conditions in order to use the normal approximation.
So we see that we satisfy the conditions and then we can apply the approximation.
If we appply the approximation the new mean and standard deviation are:
So we can approximate the random variable X like this:
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean". The letter is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words:
The z score is given by this formula:
If we replace we got:
Let's assume that we conduct the following test:
H0:
H1:
We want to check is the score for the student is significantly less than the expected value using random guessing.
So on this case since we have the statistic we can calculate the p value on this way: