Respuesta :
Answer:
i) 0.1% probability that if the coin is actually fair, we reach a false conclusion.
ii) 0.05% probability that if the coin is actually unfair, we reach a false conclusion
Step-by-step explanation:
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
Fair coin:
Comes up heads 50% of the time, so [tex]p = 0.5[/tex]
1000 trials, so [tex]n = 1000[/tex]
So
[tex]E(X) = np = 1000*0.5 = 500[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1000*0.5*0.5} = 15.81[/tex]
If the coin lands on heads 550 or more times, then we shall conclude that it is a biased coin.
(i) If the coin is actually fair, what is the probability that we shall reach a false conclusion?
This is the probability that the number of heads is 550 or more, so this is 1 subtracted by the pvalue of Z when X = 549.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{549 - 500}{15.81}[/tex]
[tex]Z = 3.1[/tex]
[tex]Z = 3.1[/tex] has a pvalue of 0.9990
1 - 0.9990 = 0.001
0.1% probability that if the coin is actually fair, we reach a false conclusion.
(ii) If the coin is actually unfair, what is the probability that we shall reach a false conclusion?
Comes up heads 60% of the time, so [tex]p = 0.6[/tex]
1000 trials, so [tex]n = 1000[/tex]
So
[tex]E(X) = np = 1000*0.6 = 600[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1000*0.6*0.4} = 15.49[/tex]
If the coin lands on less than 550 times(that is, 549 or less), then we shall conclude that it is a biased coin.
So this is the pvalue of Z when X = 549.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{549 - 600}{15.49}[/tex]
[tex]Z = -3.29[/tex]
[tex]Z = -3.29[/tex] has a pvalue of 0.0005
0.05% probability that if the coin is actually unfair, we reach a false conclusion
