Respuesta :
Answer:
0.001 = 0.1% probability that a sample of this size would have a mean of 64.82 inches or less.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
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. 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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The heights of adults in a certain town have a mean of 65.42 inches with a standard deviation of 2.32 inches.
This means that [tex]\mu = 65.42, \sigma = 2.32[/tex]
Sample of 144:
This means that [tex]n = 144, s = \frac{2.32}{\sqrt{144}} = 0.1933[/tex]
Find the probability that a sample of this size would have a mean of 64.82 inches or less.
This is the pvalue of Z when X = 64.82. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{64.82 - 65.42}{0.1933}[/tex]
[tex]Z = -3.1[/tex]
[tex]Z = -3.1[/tex] has a pvalue of 0.001
0.001 = 0.1% probability that a sample of this size would have a mean of 64.82 inches or less.
