Question

For all hypothesis tests, you must show the four steps:

1. Hypotheses

2. Test statistic

3. p-value or p-value approximation

4. Conclusion sentence (Do no just say ”Reject the null hypothesis” or ”Fail to reject the

null hypothesis”)

If doing the hypothesis test in jamovi, you must include the jamovi output but show the four

steps separately as well.

Exercises

1. The Department of Natural Resources (DNR) received a complaint from recreational fishermen 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 residing 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.

Data available online: Ex 1 Data

Above

5.2

4.8

5.1

5.0

4.9

4.8

5.0

4.7

4.7

5.0

4.7

5.1

5.0

4.9

4.9

Below

4.2

4.4

4.7

4.9

4.6

4.8

4.9

4.6

5.1

4.3

5.5

4.7

4.9

4.8

4.7

(a) Perform a hypothesis test to test whether the mean oxygen levels upstream (above) is unequal to

the mean oxygen content downstream (below). Use α = 0.01. (10 points)

(b) Estimate the size of the difference in the mean dissolved oxygen readings for the two locations on

the river using a 99% confidence interval. Assume σ1does not =σ2 (5 points)

(c) Assuming σ1=σ2, estimate the size of the difference in the mean dissolved oxygen readings for

the two locations on the river using a 99% confidence interval. (5 points)

Answer 1

For Above

= 4.92, s1 = 0.16, n1 = 15

For below

= 4.74, s2 = 0.32, n2 = 15

a) H₀:

   H₁:

The test statistic is

= 20

P-value = 2 * P(T > 1.95)

           = 2 * (1 - P(T < 1.95))

          = 2 * (1 - 0.9673)

          = 0.0654

Since the P-value is greater than the significance level(0.0654 > 0.01), so we should not reject the null hypothesis.

At 0.01 significance level, there is not sufficient evidence to conclude that the mean oxygen levels upstream (above) is unequal to the mean oxygen content downstream (below).

b) At 99% confidence level, the critical value is t^* = 2.845

The 99% confidence interval is

c) The pooled sample proportion is

     

    

DF = 15 + 15 - 2 = 28

At 99% confidence level, the critical value is t^* = 2.763

The 99% confidence interval is