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ExtremeBDS [4]
3 years ago
13

QUESTION 9 Imagine that the mean height for all Division I women's basketball programs is 69 inches with a standard deviation of

3 inches. 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? (Round z score to two decimal places.)
Mathematics
1 answer:
fenix001 [56]3 years ago
3 0

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 \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

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 \mu and standard deviation \sigma, the sample means with size n of at least 30 can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}

In this problem, we have that:

\mu = 69, \sigma = 3, n = 10, s = \frac{3}{\sqrt{10}} = 0.9487

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

Z = \frac{X - \mu}{\sigma}

By the Central limit theorem

Z = \frac{X - \mu}{s}

Z = \frac{71.2 - 69}{0.9487}

Z = 2.32

Z = 2.32 has a pvalue of 0.9898

98.98% of means would fall below that for these UConn Huskies

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Substituting x= 5 and y = 38 into equation 1, it becomes

5 + 38 + z = 80

z = 80 - 43

z = 37

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