Answer:
0.9544 = 95.44% probability that a new sample this year of 100 steady smokers spends between $19 and $21 on average
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 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.
Central Limit Theorem
The Central Limit Theorem estabilishes 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.
The firm found that the population mean amount that all smokers spend on cigarettes is $20 and the population standard deviation is $5.
This means that ![\mu = 20, \sigma = 5](https://tex.z-dn.net/?f=%5Cmu%20%3D%2020%2C%20%5Csigma%20%3D%205)
Sample of 100:
This means that ![n = 100, s = \frac{5}{\sqrt{100}} = 0.5](https://tex.z-dn.net/?f=n%20%3D%20100%2C%20s%20%3D%20%5Cfrac%7B5%7D%7B%5Csqrt%7B100%7D%7D%20%3D%200.5)
What is the probability that a new sample this year of 100 steady smokers spends between $19 and $21 on average?
This is the pvalue of Z when X = 21 subtracted by the pvalue of Z when X = 19. So
X = 21
![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{21 - 20}{0.5}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B21%20-%2020%7D%7B0.5%7D)
![Z = 2](https://tex.z-dn.net/?f=Z%20%3D%202)
has a pvalue of 0.9772
X = 19
![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{19 - 20}{0.5}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B19%20-%2020%7D%7B0.5%7D)
![Z = -2](https://tex.z-dn.net/?f=Z%20%3D%20-2)
has a pvalue of 0.0228
0.9772 - 0.0228 = 0.9544
0.9544 = 95.44% probability that a new sample this year of 100 steady smokers spends between $19 and $21 on average