Answer:
a) 0.2741 = 27.41% probability that at least 13 believe global warming is occurring
b) 0.7611 = 76.11% probability that at least 110 believe global warming is occurring
Step-by-step explanation:
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
In which is the number of different combinations of x objects from a set of n elements, given by the following formula.
And p is the probability of X happening.
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 ,
.
In this problem, we have that:
(a) For a sample of 16 Americans, what is the probability that at least 13 believe global warming is occurring?
Here , we want
. So
In which
0.2741 = 27.41% probability that at least 13 believe global warming is occurring
(b) For a sample of 160 Americans, what is the probability that at least 110 believe global warming is occurring?
Now . So
Using continuity correction, this is , which is 1 subtracted by the pvalue of Z when X = 109.5. So
has a pvalue of 0.2389
1 - 0.2389 = 0.7611
0.7611 = 76.11% probability that at least 110 believe global warming is occurring