Respuesta :
Answer:
0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject
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 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]
37% of Americans say that math is the most important subject in school.
This means that [tex]p = 0.37[/tex]
Sample of 400 Americans
This means that [tex]n = 400[/tex]
Mean and standard deviation:
[tex]\mu = p = 0.37[/tex]
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.37*0.63}{400}} = 0.0241[/tex]
What is the probability that between 40% and 45% will say that math is the most important subject?
This is the pvalue of Z when X = 0.45 subtracted by the pvalue of Z when X = 0.4. So
X = 0.45
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.45 - 0.37}{0.0241}[/tex]
[tex]Z = 3.31[/tex]
[tex]Z = 3.31[/tex] has a pvalue of 0.9995
X = 0.4
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.4 - 0.37}{0.0241}[/tex]
[tex]Z = 1.24[/tex]
[tex]Z = 1.24[/tex] has a pvalue of 0.8925
0.9995 - 0.8925 = 0.1070
0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject
