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A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 30.8 ng/mL with a standard deviation of 4.371 ng/mL. Assuming the true distribution of blood vitamin D levels follows a normal distribution, if you randomly select a landscaper in the US, what is the likelihood that his/her vitamin D level will be between 36.84 and 39.73 ng/mL

Sagot :

Answer:

0.0631 = 6.31% probability that his/her vitamin D level will be between 36.84 and 39.73 ng/mL

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

The population average level of vitamin D in US landscapers was found to be 30.8 ng/mL with a standard deviation of 4.371 ng/mL

This means that [tex]\mu = 30.8, \sigma = 4.371[/tex]

What is the likelihood that his/her vitamin D level will be between 36.84 and 39.73 ng/mL?

This is the pvalue of Z when X = 39.73 subtracted by the pvalue of Z when X = 36.84.

X = 39.73

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{39.73 - 30.8}{4.371}[/tex]

[tex]Z = 2.04[/tex]

[tex]Z = 2.04[/tex] has a pvalue of 0.9793

X = 36.84

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{36.84 - 30.8}{4.371}[/tex]

[tex]Z = 1.38[/tex]

[tex]Z = 1.38[/tex] has a pvalue of 0.9162

0.9793 - 0.9162 = 0.0631

0.0631 = 6.31% probability that his/her vitamin D level will be between 36.84 and 39.73 ng/mL

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