Respuesta :
Answer:
0.3758 = 37.58% probability that a sample mean estimate will lie within 1% of the mean of the population of the estimates of all economists.
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.
Mean 26% and standard deviation 12%.
This means that [tex]\mu = 26, \sigma = 12[/tex]
Sample of 35:
This means that [tex]n = 35, s = \frac{12}{\sqrt{35}}[/tex]
What is the approximate probability that a sample mean estimate, based on a random sample of n = 35 economists, will lie within 1% of the mean of the population of the estimates of all economists?
This is the p-value of Z when X = 26 + 1 = 27 subtracted by the p-value of Z when X = 26 - 1 = 25. So
X = 27
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{27 - 26}{\frac{12}{\sqrt{35}}}[/tex]
[tex]Z = 0.49[/tex]
[tex]Z = 0.49[/tex] has a p-value of 0.6879
X = 25
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{25 - 26}{\frac{12}{\sqrt{35}}}[/tex]
[tex]Z = -0.49[/tex]
[tex]Z = -0.49[/tex] has a p-value of 0.3121
0.6879 - 0.3121 = 0.3758
0.3758 = 37.58% probability that a sample mean estimate will lie within 1% of the mean of the population of the estimates of all economists.
