The concentration of mercury in a lake has been monitored for a number of years. Measurements taken on a weekly basis yielded an average of 1.20 mg/m3 (milligrams per cubic meter). Following an accident at a smelter on the shore of the lake, there was some concern that the mean mercury concentration might have increased and 15 measurements of mercury concentrations were taken. Assuming that the standard deviation of the mercury concentration is. 32 mg/m3, calculate the power of the test to detect mercury concentration of 1.48 mg/m3.

Step-by-step explanation:

Let us set up the hypothesis.

For the Null Hypothesis,

H0: μ = 1.2

For the alternative hypothesis, it includes only values greater than that of the null hypothesis. Since we are finding the power of this test for a specific value of 1.48 mg/m3, then the

Alternative Hypothesis would be

H1: μ = 1.48

Let us assume a level of significance of 5%. The z score from the normal distribution table is 1.645. This means that we would reject H0 if the z score calculated from the test is greater than 1.645. To calculate the z score, we would apply the formula,

z = (x - µ) / (σ/√n)

Where

x = sample mean

µ = population mean

σ = standard deviation

n = number of samples

From the information given,

µ = 1.2

σ = 0.32

n = 15

Therefore,

z = (x - 1.2) / (0.32/√15)

Substituting the z critical value of 1.645, it becomes

1.645 = (x - 1.2) / (0.32/√15)

x - 1.2 = 0.08262364472 * 1.645

x = 1.2 + 0.14 = 1.34

If we assume the alternative hypothesis to be true, then μ = 1.48

Calculating the z statistic, it becomes

z = (x - 1.48) / (0.32/√15)

z = (1.34 - 1.48) / (0.32/√15) = - 1.69

From the normal distribution table,

P (z > - 1.69) is 0.95

The power of the test is 0.95 * 100 = 95%. It is high because the power of statistically powerful tests is 80% and above.