Answer:
a,b) x represents the general attitude of these students toward recreational reading.
c) The 10th percentile of sample means is 102.51.
d) ![P(x < 100) = 0.01390](https://tex.z-dn.net/?f=P%28x%20%3C%20100%29%20%3D%200.01390)
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)
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 This p-value is the probability that the value of the measure is greater than X.
In this case, we have that:
The mean score for this population of children was 106 points, with a standard deviation of 16.4 points, so
.
a,b) What is x?
x represent the mean recreational reading attitude score for the sample. So x represents the general attitude of these students toward recreational reading.
c) What sample mean would be the cutoff for the bottom 10% of sample means. (You are being asked for the 10th percentile of sample means.)
This is the value of X when Z has a pvalue of 0.10.
Looking at the Z table, that is
.
We are working with the mean of the sample, so we have to find the standard deviation of the sample. That is
![s = \frac{\sigma}{\sqrt{n}} = \frac{16.4}{\sqrt{36}} = 2.73](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%20%3D%20%5Cfrac%7B16.4%7D%7B%5Csqrt%7B36%7D%7D%20%3D%202.73)
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-1.28 = \frac{X - 106}{2.73}](https://tex.z-dn.net/?f=-1.28%20%3D%20%5Cfrac%7BX%20-%20106%7D%7B2.73%7D)
![X - 106 = -1.28*2.73](https://tex.z-dn.net/?f=X%20-%20106%20%3D%20-1.28%2A2.73)
![X = 102.51](https://tex.z-dn.net/?f=X%20%3D%20102.51)
The 10th percentile of sample means is 102.51.
d) Find P(x < 100).
This is the pvalue of Z when X = 100.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{100 - 106}{2.73}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B100%20-%20106%7D%7B2.73%7D)
![Z = -2.20](https://tex.z-dn.net/?f=Z%20%3D%20-2.20)
has a pvalue of 0.01390.
So
.