Respuesta :
Answer:
0.9173 = 91.73% probability that the average duration of unemployment was between 30 and 37 weeks.
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 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.
Mean of 34.6, standard deviation of 10.2
This means that [tex]\mu = 34.6, \sigma = 10.2[/tex]
Sample of 36
This means that [tex]n = 36, s = \frac{10.2}{\sqrt{36}} = 1.7[/tex]
What is the probability that the average duration of unemployment was between 30 and 37 weeks?
This is the pvalue of Z when X = 37 subtracted by the pvalue of Z when X = 30.
X = 37
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{37 - 34.6}{1.7}[/tex]
[tex]Z = 1.41[/tex]
[tex]Z = 1.41[/tex] has a pvalue of 0.9207
X = 30
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{30 - 34.6}{1.7}[/tex]
[tex]Z = -2.71[/tex]
[tex]Z = -2.71[/tex] has a pvalue of 0.0034
0.9207 - 0.0034 = 0.9173
0.9173 = 91.73% probability that the average duration of unemployment was between 30 and 37 weeks.
