1. Your company is comparing the ammonium (NH₄⁺) concentrations in rainfall between an industrial site and a residential site. You analyzed the data and found that a test is unable to distinguish any difference between the two groups of samples (t=0.03, p>0.98, two-tailed). Conversely, a rank-sum test does show a difference (U=76.5, p slightly greater than 0.05), albeit with a slightly liberal interpretation of the typically desired 5% false positive rate. Following standard protocols at your firm, you do a quality control check on the data and notice to your horror that the farthest outlier at the residential site has a coding error (and it’s a big one): Instead of 9.7 ppm, it should have been 0.97 ppm. Now, with the coding error corrected, the concentrations (in ppm) are as follows:| Residential site | Industrial site ||------------------|-----------------|| 0.30 | 0.59 || 0.36 | 0.87 || 0.50 | 1.1 || 0.70 | 1.1 || 0.70 | 1.2 || 0.90 | 1.3 || 0.92 | 1.6 || 0.97 | 1.7 || 1.0 | 3.2 || 1.3 | 4.0 |Let’s compare results from a t-test with results from a rank-sum test, now that the error has been corrected.(a) What are the means and variances of the two groups of ammonium concentrations? Are the variances similar enough that they can be pooled?(b) What is the standard error of the difference between means? How many degrees of freedom are there?(c) What is the value of t, and what is the statistical significance of this value (for a two-tailed test)?(d) Did fixing the mistake in the outlier have a large effect on the difference between the means? Did it have a large effect on the statistical significance of the difference?(e) Calculate the Mann-Whitney statistics U and U' and, using the “exact test” (i.e., by looking up the critical

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(a) What are the means and variances of the two groups of ammonium concentrations? Are the variances similar enough that they can be pooled? (b) What is the standard error of the difference between means? How many degrees of freedom are there? (c) What is the value of t, and what is the statistical significance of this value (for a two-tailed test)?

(a) What are the means and variances of the two groups of ammonium concentrations? Are the variances similar enough that they can be pooled? (b) What is the standard error of the difference between means? How many degrees of freedom are there? (c) What is the value of t, and what is the statistical significance of this value (for a two-tailed test)?

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.

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Question

1. Your company is comparing the ammonium (NH₄⁺) concentrations in rainfall between an industrial site and a residential site. You analyzed the data and found that a test is unable to distinguish any difference between the two groups of samples (t=0.03, p>0.98, two-tailed). Conversely, a rank-sum test does show a difference (U=76.5, p slightly greater than 0.05), albeit with a slightly liberal interpretation of the typically desired 5% false positive rate. Following standard protocols at your firm, you do a quality control check on the data and notice to your horror that the farthest outlier at the residential site has a coding error (and it’s a big one): Instead of 9.7 ppm, it should have been 0.97 ppm. Now, with the coding error corrected, the concentrations (in ppm) are as follows:

| Residential site | Industrial site |
|------------------|-----------------|
| 0.30 | 0.59 |
| 0.36 | 0.87 |
| 0.50 | 1.1 |
| 0.70 | 1.1 |
| 0.70 | 1.2 |
| 0.90 | 1.3 |
| 0.92 | 1.6 |
| 0.97 | 1.7 |
| 1.0 | 3.2 |
| 1.3 | 4.0 |

Let’s compare results from a t-test with results from a rank-sum test, now that the error has been corrected.

(a) What are the means and variances of the two groups of ammonium concentrations? Are the variances similar enough that they can be pooled?

(b) What is the standard error of the difference between means? How many degrees of freedom are there?

(c) What is the value of t, and what is the statistical significance of this value (for a two-tailed test)?

(d) Did fixing the mistake in the outlier have a large effect on the difference between the means? Did it have a large effect on the statistical significance of the difference?

(e) Calculate the Mann-Whitney statistics U and U' and, using the “exact test” (i.e., by looking up the critical

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