Lake Macatawa Bacteria: 90% Confidence Analysis

Chapter 7, Problem 5

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QUESTION:

As a clue to the amount of organic waste in Lake Macatawa (see Example 7.1-4), a count was made of the number of bacteria colonies in 100 milliliters of water. The number of colonies, in hundreds, for \(n = 30\) samples of water from the east basin yielded

\(\begin{array}{rrrrrrrrrr} 93 & 140 & 8 & 120 & 3 & 120 & 33 & 70 & 91 & 61 \\ 7 & 100 & 19 & 98 & 110 & 23 & 14 & 94 & 57 & 9 \\ 66 & 53 & 28 & 76 & 58 & 9 & 73 & 49 & 37 & 92 \end{array}\)

Find an approximate 90% confidence interval for the mean number (say, \(\mu_{E}\)) of colonies in 100 milliliters of water in the east basin

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QUESTION:

As a clue to the amount of organic waste in Lake Macatawa (see Example 7.1-4), a count was made of the number of bacteria colonies in 100 milliliters of water. The number of colonies, in hundreds, for \(n = 30\) samples of water from the east basin yielded

\(\begin{array}{rrrrrrrrrr} 93 & 140 & 8 & 120 & 3 & 120 & 33 & 70 & 91 & 61 \\ 7 & 100 & 19 & 98 & 110 & 23 & 14 & 94 & 57 & 9 \\ 66 & 53 & 28 & 76 & 58 & 9 & 73 & 49 & 37 & 92 \end{array}\)

Find an approximate 90% confidence interval for the mean number (say, \(\mu_{E}\)) of colonies in 100 milliliters of water in the east basin

ANSWER:

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The  confidence interval for the mean number of colonies can be computed using the formula,

\( CI = \overline x  \pm {t_{\left( {n - 1} \right),\alpha /2}}\frac{s}{{\sqrt n }} \)

Here, \( \overline x \)  is the mean number of colonies for the sample, n is the sample size, \(\sigma \) is the standard deviation, and \( {t_{\left( {n - 1} \right),\alpha /2}} \) the critical value for the confidence interval at \(\alpha \) significance level and n-1 degrees of freedom. The two-tailed critical value should be taken into consideration.

Therefore, the sample mean of the bacteria colonies can be calculated as follows:

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Lake Macatawa Bacteria: 90% Confidence Analysis
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Discover the method to analyze bacteria colonies in Lake Macatawa's east basin. Using 30 samples, we compute a 90% confidence interval for the mean colony count. Learn the steps of statistical analysis in environmental studies.


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