The carbon dioxide emissions of a group of nations had a mean of 8.7 and standard deviation of 2.1. a. One country's observation was 14.1. Find its z-score and interpret. b. Another country's observation was 2.2. Find its z-score and interpret. c. A third country's observation was 9.1. Find its z-score and interpret.

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27-02-2020
Mathematics

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The carbon dioxide emissions of a group of nations had a mean of 8.7 and standard deviation of 2.1. a. One country's observation was 14.1. Find its z-score and interpret. b. Another country's observation was 2.2. Find its z-score and interpret. c. A third country's observation was 9.1. Find its z-score and interpret.

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joaobezerra joaobezerra · Answer 1 · 2020-02-27T01:48:59+01:00

Answer:

a) [tex]Z = 2.57[/tex]

This country emmits 2.57 standard deviations above the mean of the emissions of the countries of this group of nations.

b) [tex]Z = -3.1[/tex]

This country emmits 3.1 standard deviations below the mean of the emissions of the countries of this group of nations.

c) [tex]Z = 0.19[/tex]

This country emmits 0.19 standard deviations above the mean of the emissions of the countries of this group of nations.

Step-by-step explanation:

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.

In this problem, we have that:

[tex]\mu = 8.7, \sigma = 2.1[/tex]

a. One country's observation was 14.1. Find its z-score and interpret.

Here we have [tex]X = 14.1[/tex]

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

[tex]Z = \frac{14.1 - 8.7}{2.1}[/tex]

[tex]Z = 2.57[/tex]

This country emmits 2.57 standard deviations above the mean of the emissions of the countries of this group of nations.

b. Another country's observation was 2.2. Find its z-score and interpret.

Here we have [tex]X = 2.2[/tex]

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

[tex]Z = \frac{2.2 - 8.7}{2.1}[/tex]

[tex]Z = -3.1[/tex]

This country emmits 3.1 standard deviations below the mean of the emissions of the countries of this group of nations.

c. A third country's observation was 9.1. Find its z-score and interpret.

Here we have [tex]X = 9.1[/tex]

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

[tex]Z = \frac{9.1 - 8.7}{2.1}[/tex]

[tex]Z = 0.19[/tex]

This country emmits 0.19 standard deviations above the mean of the emissions of the countries of this group of nations.