Answer:
35% of all sample proportions be will be above 0.6074.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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.
In this problem, we have that:
Of the 1,300 children participating in a town's parks and recreations programs, 765 are under the age of 8. This means that ![\mu = \frac{765}{1300} = 0.5885](https://tex.z-dn.net/?f=%5Cmu%20%3D%20%5Cfrac%7B765%7D%7B1300%7D%20%3D%200.5885)
In the sampling distribution of sample proportions of size 100, above what proportion will 35% of all sample proportions be?
We are using sample proportions of size 100, so
.
In the sampling distribution of sample proportions of size 100, above what proportion will 35% of all sample proportions be?
This proportion is the value of X in the 65th percentile, that is, when Z has a pvalue of 0.65. This is
. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![0.385 = \frac{X - 0.5885}{0.0492}](https://tex.z-dn.net/?f=0.385%20%3D%20%5Cfrac%7BX%20-%200.5885%7D%7B0.0492%7D)
![X - 0.5885 = 0.385*0.0492](https://tex.z-dn.net/?f=X%20-%200.5885%20%3D%200.385%2A0.0492)
![X = 0.6074](https://tex.z-dn.net/?f=X%20%3D%200.6074)
35% of all sample proportions be will be above 0.6074.