Answer:
a) 0.259
b) 0.297
c) 0.497
Step-by-step explanation:
To solve this problem, it is important to know 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 random variable X, with mean
and standard deviation
, a large sample size can be approximated to a normal distribution with mean
and standard deviation ![s = \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
In this problem, we have that:
![\mu = 11, \sigma = 3](https://tex.z-dn.net/?f=%5Cmu%20%3D%2011%2C%20%5Csigma%20%3D%203)
a. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?
Here we have that ![n = 25, s = \frac{3}{\sqrt{25}} = 0.6](https://tex.z-dn.net/?f=n%20%3D%2025%2C%20s%20%3D%20%5Cfrac%7B3%7D%7B%5Csqrt%7B25%7D%7D%20%3D%200.6)
This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.
X = 11.2
![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{11.2 - 11}{0.6}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B11.2%20-%2011%7D%7B0.6%7D)
![Z = 0.33](https://tex.z-dn.net/?f=Z%20%3D%200.33)
has a pvalue of 0.6293.
X = 10.8
![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{10.8 - 11}{0.6}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B10.8%20-%2011%7D%7B0.6%7D)
![Z = -0.33](https://tex.z-dn.net/?f=Z%20%3D%20-0.33)
has a pvalue of 0.3707.
0.6293 - 0.3707 = 0.2586
0.259 probability, rounded to three decimal places.
b. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.5 and 11 minutes?
Subtraction of the pvalue of Z when X = 11 subtracted by the pvalue of Z when X = 10.5. So
X = 11
![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{11 - 11}{0.6}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B11%20-%2011%7D%7B0.6%7D)
![Z = 0](https://tex.z-dn.net/?f=Z%20%3D%200)
has a pvalue of 0.5.
X = 10.5
![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{10.5 - 11}{0.6}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B10.5%20-%2011%7D%7B0.6%7D)
![Z = -0.83](https://tex.z-dn.net/?f=Z%20%3D%20-0.83)
has a pvalue of 0.2033.
0.5 - 0.2033 = 0.2967
0.297, rounded to three decimal places.
c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?
Here we have that ![n = 100, s = \frac{3}{\sqrt{100}} = 0.3](https://tex.z-dn.net/?f=n%20%3D%20100%2C%20s%20%3D%20%5Cfrac%7B3%7D%7B%5Csqrt%7B100%7D%7D%20%3D%200.3)
This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.
X = 11.2
![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{11.2 - 11}{0.3}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B11.2%20-%2011%7D%7B0.3%7D)
![Z = 0.67](https://tex.z-dn.net/?f=Z%20%3D%200.67)
has a pvalue of 0.7486.
X = 10.8
![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{10.8 - 11}{0.3}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B10.8%20-%2011%7D%7B0.3%7D)
![Z = -0.67](https://tex.z-dn.net/?f=Z%20%3D%20-0.67)
has a pvalue of 0.2514.
0.7486 - 0.2514 = 0.4972
0.497, rounded to three decimal places.