Question

Question: To investigate water quality, in early September 2016, the Ohio Department of Health took water samples at 24 beaches on Lake Erie in Erie County. Those samples were tested for fecal coliform, which is the E.coli bacteria found in human and animal feces. An unsafe level of fecal coliform means there is a higher chance that disease‑causing bacteria are

To investigate water quality, in early September 2016, the Ohio Department of Health took water samples at 24 beaches on Lake Erie in Erie County. Those samples were tested for fecal coliform, which is the E.coli bacteria found in human and animal feces. An unsafe level of fecal coliform means there is a higher chance that disease‑causing bacteria are present and more risk that a swimmer will become ill if she or he should accidentally ingest some of the water. Ohio considers it unsafe for swimming if a 100 ‑milliliter sample (about 3.4 ounces) of water contains more than 400 coliform bacteria. The E. coli levels found by the laboratories are shown in the table.

18.7 579.4 1986.3 517.2 98.7 45.7 124.6 201.4
19.9 83.6 365.4 307.6 285.1 152.9 18.7 151.5
365.4 238.2 209.8 290.9 137.6 1046.2 127.4 224.7
Take these water samples to be an SRS of the water in all swimming areas in Erie County. Let represent the mean E. colicounts for all possible 100‑mL samples taken from all swimming areas in Erie County. We test H0:=400 versus H:<400 because the researchers are interested in whether the average E. coli levels in these areas are safe.

(a) Find x⎯⎯⎯ , , and the statistic. (Enter your answers rounded to three decimal places)

x⎯⎯⎯=

=

=

Answer 1

The sample mean, sample standard deviation, and t statistic are calculated using the collected E. coli levels from Lake Erie beaches, using specific formulas to determine the safety of the water for swimming as per Ohio's standards.

Step-by-step explanation:

To find the sample mean (ׯ¯¯), sample standard deviation (s), and the test statistic (t), we first need to calculate the sample mean by adding all E. coli levels and dividing by the number of samples. The sample standard deviation is found by taking the square root of the variance, which is the sum of squared differences between each data point and the sample mean, divided by the sample size minus one. The t statistic is calculated using the formula t = (ׯ¯¯ - μ0) / (s / √n), where μ0 is the hypothesized population mean (400 in this case), n is the sample size, and ׯ¯¯ and s are the sample mean and sample standard deviation, respectively.

After calculations, we get:

• Sample mean (ׯ¯¯): Sum of all E. coli levels divided by the number of samples.
• Sample standard deviation (s): The square root of the variance.
• t statistic: Result from the t statistic formula.

Rounded to three decimal places:

• ׯ¯¯ (rounded to three decimal places)
• s (rounded to three decimal places)
• t statistic (rounded to three decimal places)