Respuesta :
Answer:
c) Find the area to the left of z = -1.2 under a standard normal curve.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probabiliy distribution
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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, or the area to the left of Z in the normal curve. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X, which is the area to the right of Z in the normal curve.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sample means with size n of at least 30 can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 65.6, \sigma = 12.5, n = 100, s = \frac{12.5}{\sqrt{100}} = 1.25[/tex]
Which of the following corretly describe how to find the probability that you obtain a sample mean age that is younger than 64 years?
Area to the left of z when X = 64 under the standard normal curve. We have to find Z.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Applying the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{64 - 65.6}{1.25}[/tex]
[tex]Z = -1.2[/tex]
So the correct answer is:
c) Find the area to the left of z = -1.2 under a standard normal curve.
