Respuesta :
Answer:
99.72% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use 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, 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]
In this question:
[tex]n = 860, p = 0.41[/tex]
So
[tex]\mu = 0.41, s = \sqrt{\frac{0.41*0.59}{860}} = 0.0168[/tex]
What is the probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%?
This is the pvalue of Z when X = 0.41 + 0.05 = 0.46 subtracted by the pvalue of Z when X = 0.41 - 0.05 = 0.36. So
X = 0.46
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.46 - 0.41}{0.0168}[/tex]
[tex]Z = 2.98[/tex]
[tex]Z = 2.98[/tex] has a pvalue of 0.9986
X = 0.36
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.36 - 0.41}{0.0168}[/tex]
[tex]Z = -2.98[/tex]
[tex]Z = -2.98[/tex] has a pvalue of 0.0014
0.9986 - 0.0014 = 0.9972
99.72% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%.
