Answer:
44.23% probability that the mean weight loss after one week on this pill for a random sample of 55 individuals will be 1.77 pounds or more
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 normally distributed samples are 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.
In this problem, we have that:
![\mu = 1.75, \sigma = 1.02, n = 55, s = \frac{1.02}{\sqrt{55}} = 0.1375](https://tex.z-dn.net/?f=%5Cmu%20%3D%201.75%2C%20%5Csigma%20%3D%201.02%2C%20n%20%3D%2055%2C%20s%20%3D%20%5Cfrac%7B1.02%7D%7B%5Csqrt%7B55%7D%7D%20%3D%200.1375)
What is the probability that the mean weight loss after one week on this pill for a random sample of 55 individuals will be 1.77 pounds or more
This is 1 subtracted by the pvalue of Z when X = 1.77. 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)
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{1.77 - 1.75}{0.1375}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B1.77%20-%201.75%7D%7B0.1375%7D)
![Z = 0.145](https://tex.z-dn.net/?f=Z%20%3D%200.145)
has a pvalue of 0.5577
1 - 0.5577 = 0.4423
44.23% probability that the mean weight loss after one week on this pill for a random sample of 55 individuals will be 1.77 pounds or more