Calculate the test statistic

Question

Biologists researching the effects of adding limestone sand as buffer for acid rain effects in streams monitored the pH levels of two streams each month for 36 months. The first stream had a mean pH level of 6.8 with a standard deviation of 2.3. The control stream had a mean pH level of 9.2 with a standard deviation of 1.5. Assume a 0.05 significance level for testing the claim that the mean pH of the first stream was less (more acidic) than the mean pH of the control stream. Also, assume the two samples are independent simple random samples selected from normally distributed populations.

Calculate test statistic

Answer 1

From the given assumptions, we know that we're going to need to use one of two possible versions of the two independent samples t-test to solve this problem:

1. The version that assumes the two population variances are the same.
2. The version that assumes the two population variances are different.

To determine which of the two versions we'll need to choose, we'll perform an F test at the 0.05 significance level to test the following hypotheses:

H_{0}: \sigma_{1} = \sigma_{2}

H_{1}: \sigma_{1} \neq \sigma_{2}

Use of the 2-SampFTest function on a TI-84 Plus CE calculator, which carries out the desired F test, then tells us that

F_{35,35} = 2.35

p-value = 0.0133 < 0.05,

which implies that the two population variances are different. Thus, we will use the version of the two independent samples t-test that assumes unequal population variances to solve our original problem. If we let

\mu_{1} = population mean pH for the first stream

\mu_{2} = population mean pH for the control stream,

then the two hypotheses we're interested in testing at the 0.05 significance level will be

H_{0}: \mu_{1} \geq \mu_{2}

H_{1}: \mu_{1} < \mu_{2}.

Use of the 2-SampTTest function on a TI-84 Plus CE calculator, which enable us to carry out the desired version of the two independent samples t-test, then tells us that our desired test statistic is

t_{60.2121247} = -5.24.