6.6 The Department of Natural Resources (DNR) received a complaint from recreational fish- ermen that a community was releasing sewage into the river where they fished. These types of releases lower the level of dissolved oxygen in the river and hence cause damage to the fish resid- ing in the river. An inspector from the DNR designs a study to investigate the fishermen’s claim. Fifteen water samples are selected at locations on the river upstream from the community and fifteen samples are selected downstream from the community. The dissolved oxygen readings in parts per million (ppm) are given in the following table.Upstream 5.2 4.8 5.1 5.0 4.9 4.8 5.0 4.7 4.7 5.0 4.6 5.2 5.0 4.9 4.7Downstream 3.2 3.4 3.7 3.9 3.6 3.8 3.9 3.6 4.1 3.3 4.5 3.7 3.9 3.8 3.7a. In order for the discharge to have an impact on fish health, there needs to be at least an .5 ppm reduction in the dissolved oxygen. Do the data provide sufficient evidence that there is a large enough reduction in the mean dissolved oxygen be- tween the upstream and downstream water in the river to impact the health of the fish? Use a 5 .01.b. Do the required conditions to use the test in part (a) appear to be valid?c. What is the level of significance of the test in part (a)?d. Estimate the size of the difference in the mean dissolved oxygen readings for the two locations on the river using a 99% confidence interval.

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Asked Feb 26, 2019
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6.6 

The Department of Natural Resources (DNR) received a complaint from recreational fish- ermen that a community was releasing sewage into the river where they fished. These types of releases lower the level of dissolved oxygen in the river and hence cause damage to the fish resid- ing in the river. An inspector from the DNR designs a study to investigate the fishermen’s claim. Fifteen water samples are selected at locations on the river upstream from the community and fifteen samples are selected downstream from the community. The dissolved oxygen readings in parts per million (ppm) are given in the following table.

Upstream 5.2 4.8 5.1 5.0 4.9 4.8 5.0 4.7 4.7 5.0 4.6 5.2 5.0 4.9 4.7

Downstream 3.2 3.4 3.7 3.9 3.6 3.8 3.9 3.6 4.1 3.3 4.5 3.7 3.9 3.8 3.7

a. In order for the discharge to have an impact on fish health, there needs to be at least an .5 ppm reduction in the dissolved oxygen. Do the data provide sufficient evidence that there is a large enough reduction in the mean dissolved oxygen be- tween the upstream and downstream water in the river to impact the health of the fish? Use a 5 .01.

b. Do the required conditions to use the test in part (a) appear to be valid?

c. What is the level of significance of the test in part (a)?

d. Estimate the size of the difference in the mean dissolved oxygen readings for the two locations on the river using a 99% confidence interval.

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Expert Answer

Step 1

Introduction to paired t-test:

The paired t-test or the paired sample t-test or the dependent sample t-test is a statistical method for testing hypotheses regarding the mean difference between two sets of observations, taken from the same individuals before and after a treatment, or between different individuals who are interdependent, leading to “pairs of observations”, instead of simply two independent sets of observations. The test is conducted using the difference between the two sets of observations.

In order to use the paired t-test, the following conditions must be satisfied:

  • The difference must be continuous in nature.
  • The sampled individuals must be independent of each other.
  • The difference must be at least approximately normally distributed.
  • The difference must not contain outliers.
Step 2

Hypotheses and level of significance:

Denote μ1 and μ2 as the mean dissolved oxygen levels (ppm) upstream and downstream respectively. Thus, the reduction in mean dissolved oxygen level is μd = μ1μ2.

The hypotheses to be tested are:

Null hypothesis:

H0: The reduction in oxygen is by less than 0.5 ppm, that is, μd < 0.5.

Alternative hypothesis:

H1: The reduction in oxygen is by at least 0.5 ppm, that is, μd ≥ 0.5.

The level of significance is given as α = 0.01, in the last line of the question in part (a).

Step 3

Calculation for part (a):

The paired t-test can be conducted using MINITAB software as follows:

Software procedure:

  • Enter the Upstream and Downstream data in two columns in a MINITAB worksheet.
  • Select Stat > Basic Statistics > Paired t
  • Choose Each sample is in a column, select Upstream in Sample 1 and Downstream in Sample 2.
  • Choose Options, enter Confidence level as 0 (as level of significance is 0.01), Hypothesized difference as 0.5, Alternative hypothesis as D...
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