Respuesta :
Answer:
Applying the Central Limit Theorem for proportions, P( phat > 0.3) = 0.3897
Step-by-step explanation:
Normal probability distribution
Problems of normally distributed samples are 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.
Central Limit Theorem
The Central Limit Theorem estabilishes 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, we have that [tex]\mu = p, s = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this problem, we have that:
[tex]p = 0.29, \mu = 0.29, n = 160, s = \sqrt{\frac{0.29*0.71}{160}} = 0.0359[/tex]
P( phat > 0.3)
This is 1 subtracted by the pvalue of Z when X = 0.3. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.3 - 0.29}{0.0359}[/tex]
[tex]Z = 0.28[/tex]
[tex]Z = 0.28[/tex] has a pvalue of 0.6103
1 - 0.6103 = 0.3897
Applying the Central Limit Theorem for proportions, P( phat > 0.3) = 0.3897
