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dchirunga23
20.03.2020 •
Mathematics
Use normal approximation to estimate the probability of passing a true/false test of 50 questions if the minimum passing grade is 60% and all responses are random guesses.
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Ответ:
7.93% probability of passing the test.
Step-by-step explanation:
We use the binomial approximation to the normal to solve this question.
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:
The standard deviation of the binomial distribution is:
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
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
,
.
Guessed true/false question.
Each have two options, one of which is correct, so![p = \frac{1}{2} = 0.5](/tpl/images/0556/5060/3e33c.png)
50 questions
This means that![n = 50](/tpl/images/0556/5060/0f144.png)
So
Probability of passing a true/false test of 50 questions if the minimum passing grade is 60% and all responses are random guesses.
This probability is 1 subtracted by the pvalue of Z when X = 0.6*50 = 30. So
1 - 0.9207 = 0.0793
7.93% probability of passing the test.
Ответ: