Pls help with some halfof problems
2 answers:
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Answer:
Since both and
, the necessary conditions are satisfied.
0.9945 = 99.45% probability that at most 85 out of 136 DVDs will work correctly.
Step-by-step explanation:
Test if the normal curve can be used as an approximation to the binomial probability by verifying the necessary conditions.
It is needed that:
and
Binomial probability distribution
Probability of exactly x successes 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 distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score 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 p-value, 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 ,
.
Assume the probability that a given DVD will work correctly is 52%.
This means that
136 DVDs
This means that
Test the conditions:
Since both and
, the necessary conditions are satisfied.
Mean and standard deviation:
Consider the probability that at most 85 out of 136 DVDs will work correctly.
Using continuity correction, this is , which is the p-value of Z when X = 85.5. So
has a p-value of 0.9945.
0.9945 = 99.45% probability that at most 85 out of 136 DVDs will work correctly.