Question

The concentration of benzene was measured in milligrams per liter for a simple random sample of five specimens of untreated wastewater produced at a gas field. The sample mean was 7.2 with a sample standard deviation of 2.8. Seven specimens of treated wastewater had an average benzene concentration of 4.2 with a standard deviation of 2.5. It is reasonable to assume that both samples come from populations that are approximately normal. Can you conclude that the mean benzene concentration differs between treated water and untreated water? Let μ1​ denote the mean benzene concentration for untreated water and μ2​ denote the mean benzene concentration for treated water. Use the p-value method with the TI-84 Plus calculator.

Answer 1

To determine if the mean benzene concentration differs between untreated and treated water, a two-sample t-test is used on a TI-84 Plus calculator. The test compares sample means using the p-value method, with a low p-value indicating a significant difference.

Step-by-step explanation:

The question involves using a TI-84 Plus calculator to determine whether there is a statistical difference between the mean concentrations of benzene in untreated and treated wastewater. To arrive at this conclusion, we'll perform a two-sample t-test to compare the means (μ1 for untreated water and μ2 for treated water) using the given sample statistics. The null hypothesis is that there is no difference between the mean concentrations of benzene in untreated and treated water (μ1 = μ2), and the alternative hypothesis is that the means are not equal (μ1 ≠ μ2).

Here's the step-by-step method to use on the TI-84 Plus:

1. Press the 'STAT' button and then navigate to 'TESTS'.
2. Select '2-SampTTest'.
3. Enter the given statistics for both samples: μ1: mean = 7.2, standard deviation = 2.8, n = 5; μ2: mean = 4.2, standard deviation = 2.5, n = 7.
4. Choose the 'Pooled' option if you assume equal variances; otherwise, choose 'No' for separate variances.
5. Set the 'Freq' to 1 for each sample, as all observations are unique.
6. Select 'Calculate' to perform the t-test.

The calculator will output a p-value. If the p-value is less than the chosen significance level (often 0.05), we reject the null hypothesis, indicating that there is a statistically significant difference between the mean concentrations of benzene in untreated and treated wastewater.