Respuesta :
Answer:
0.8665 = 86.65% probability that the sample proportion of Democrats will be greater than 52%.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Suppose 54% of the registered voters in a country are Democrats. Sample of 770.
This means that:
[tex]p = 0.54, s = \sqrt{\frac{0.54*0.46}{770}} = 0.01796[/tex]
What is the probability that the sample proportion of Democrats will be greater than 52%?
This is 1 subtracted by the p-value of Z when X = 0.52. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.52 - 0.54}{0.01796}[/tex]
[tex]Z = -1.11[/tex]
[tex]Z = -1.11[/tex] has a p-value of 0.1335
1 - 0.1335 = 0.8665
0.8665 = 86.65% probability that the sample proportion of Democrats will be greater than 52%.
