Question

An ecological group suspects dolphins are consuming dangerous levels of arsenic in a region of the world. Dangerous levels of arsenic are classified when they reach beyond 7.0 parts per million. A random sample of 28 dolphins in the region were tested and the data is shown in the stemplot.

Stemplot for Arsenic Levels.
2|4
2|
3|2 3
3|6 7 9
4|0 1 3
4|5 7 7
5|0 1 1 4 4
5|7 8
6|3 4
6|6
7|0 1 3
7|5 6 7

Key: 2|4 = 2.4 parts per million

Part A: What proportion of the dolphins in the sample have dangerous levels of arsenic in their systems? (4 points)

Part B: Construct and interpret a 99% confidence interval for the mean arsenic levels of the dolphins if the sample of 28 dolphins has a mean arsenic level of 5.26 parts per million and standard deviation of 1.49 parts per million. (6 points)

Answer 1

Part A- The proportion of dolphins in the sample with dangerous levels of arsenic is 0.

Part B- We can be 99% confident that the true mean arsenic level of the dolphins in the region lies between 4.337 and 6.183 parts per million.

Part A:

To calculate the proportion of dolphins with dangerous levels of arsenic, we need to count the number of dolphins with arsenic levels greater than 7.0 parts per million and divide by the total number of dolphins in the sample.

From the stemplot, we see that there are no dolphins with arsenic levels greater than 7.0 parts per million. Therefore, the proportion of dolphins in the sample with dangerous levels of arsenic is 0.

Answer: 0

Part B:

To construct a 99% confidence interval for the mean arsenic levels of the dolphins, we will use the following formula:

CI = X̄ ± Zα/2 * σ/√n

where:

X̄ = sample mean arsenic level = 5.26 parts per million

Zα/2 = critical value for a 99% confidence level = 2.576 (from the standard normal distribution)

σ = sample standard deviation = 1.49 parts per million

n = sample size = 28

Substituting the values, we get:

CI = 5.26 ± 2.576 * 1.49/√28

= 5.26 ± 0.923

= (4.337, 6.183)

Therefore, we can be 99% confident that the true mean arsenic level of the dolphins in the region lies between 4.337 and 6.183 parts per million.

Interpretation: If we were to take multiple random samples of 28 dolphins from the region and calculate the mean arsenic level for each sample, 99% of the intervals calculated using this formula would contain the true mean arsenic level of the population.

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