Answer:
The z-score for this data is Z = -0.26.
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
and standard deviation
, the z-score 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 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
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
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
and standard deviation ![s = \sqrt{\frac{p(1-p)}{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D)
This is based on knowledge that in the U.S., the mean PAL is 1.65 and the standard deviation is 0.55.
This means that ![\mu = 1.65, \sigma = 0.55](https://tex.z-dn.net/?f=%5Cmu%20%3D%201.65%2C%20%5Csigma%20%3D%200.55)
A study took a random sample of 51 people who lived in low income neighborhoods and found their mean PAL to be 1.63.
This means that ![n = 51, X = 1.63](https://tex.z-dn.net/?f=n%20%3D%2051%2C%20X%20%3D%201.63)
Using a one-sample z test, what is the z-score for this data
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{1.63 - 1.65}{\frac{0.55}{\sqrt{51}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B1.63%20-%201.65%7D%7B%5Cfrac%7B0.55%7D%7B%5Csqrt%7B51%7D%7D%7D)
![Z = -0.26](https://tex.z-dn.net/?f=Z%20%3D%20-0.26)
The z-score for this data is Z = -0.26.