Respuesta :
Answer:
98.98% of means would fall below that for these UConn Huskies
Step-by-step explanation:
To solve this qustion, 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 [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 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 = 69, \sigma = 3, n = 10, s = \frac{3}{\sqrt{10}} = 0.9487[/tex]
The 2010–2011 women's basketball team at the University of Connecticut, with 10 players listed on the roster, had an average height of 71.2 inches. Using the z statistic, what percent of means would fall below that for these UConn Huskies?
This is the pvalue of Z when X = 71.2. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central limit theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{71.2 - 69}{0.9487}[/tex]
[tex]Z = 2.32[/tex]
[tex]Z = 2.32[/tex] has a pvalue of 0.9898
98.98% of means would fall below that for these UConn Huskies
