Answer:
9.01% probability that less than half the sample will say they support the increase
Step-by-step explanation:
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.
For a proportion p in a sample of size n, we have that [tex]\mu = p, \sigma = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this problem
[tex]\mu = 0.53, \sigma = \sqrt{\frac{0.53*0.47}{500}} = 0.0223[/tex]
What is the probability that less than half the sample will say they support the increase?
This is the pvalue of Z when X = 0.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{0.5 - 0.53}{0.0223}[/tex]
[tex]Z = -1.34[/tex]
[tex]Z = -1.34[/tex] has a pvalue of 0.0901
9.01% probability that less than half the sample will say they support the increase
