Question

Analyses of drinking water samples for 100 homes in each of two different sections of a city gave the following means and standard deviations of lead levels (in parts per million).

  

	section 1	section 2
sample size	100	100
mean	34.5	36.2
standard deviation	5.8	6.0

(a) Calculate the test statistic and its p-value to test for a difference in the two population means. (Use Section 1 − Section 2. Round your test statistic to two decimal places and your p-value to four decimal places.)

z =

p-value =

Use the p-value to evaluate the statistical significance of the results at the 5% level.

a. H0 is not rejected. There is sufficient evidence to indicate a difference in the mean lead levels for the two sections of the city.

b. H0 is rejected. There is sufficient evidence to indicate a difference in the mean lead levels for the two sections of the city.   

c. H0 is rejected. There is insufficient evidence to indicate a difference in the mean lead levels for the two sections of the city.

d. H0 is not rejected. There is insufficient evidence to indicate a difference in the mean lead levels for the two sections of the city.

(b) Calculate a 95% confidence interval to estimate the difference in the mean lead levels in parts per million for the two sections of the city. (Use Section 1 − Section 2. Round your answers to two decimal places.)

parts per million______ to________parts per million

(c) Suppose that the city environmental engineers will be concerned only if they detect a difference of more than 5 parts per million in the two sections of the city. Based on your confidence interval in part (b), is the statistical significance in part (a) of practical significance to the city engineers? Explain.

a. Since all of the probable values of μ1 − μ2 given by the interval are all less than −5, it is likely that the difference will be more than 5 ppm, and hence the statistical significance of the difference is of practical importance to the the engineers.

b. Since all of the probable values of μ1 − μ2 given by the interval are all greater than 5, it is likely that the difference will be more than 5 ppm, and hence the statistical significance of the difference is of practical importance to the the engineers.   

c. Since all of the probable values of μ1 − μ2 given by the interval are between −5 and 5, it is not likely that the difference will be more than 5 ppm, and hence the statistical significance of the difference is not of practical importance to the the engineers.

Answer 1

The statistical software output for this problem is:

Two sample Z summary hypothesis test:
μ₁ : Mean of population 1 (Std. dev. = 5.8)
μ₂ : Mean of population 2 (Std. dev. = 6)
μ₁ - μ₂ : Difference between two means
H₀ : μ₁ - μ₂ = 0
H_A : μ₁ - μ₂ ≠ 0

Hypothesis test results:

Difference	n1	n2	Sample mean	Std. err.	Z-stat	P-value
μ1 - μ2	100	100	-1.7	0.83450584	-2.0371337	0.0416

95% confidence interval results:

Difference	n1	n2	Sample mean	Std. err.	L. limit	U. limit
μ1 - μ2	100	100	-1.7	0.83450584	-3.3356014	-0.064398607

Hence,

a) z = -2.04

P - value = 0.0416

H0 is rejected. There is sufficient evidence to indicate a difference in the mean lead levels for the two sections of the city. Option B is correct.

b) 95% confidence interval:

-3.34 to -0.06

Since all of the probable values of μ1 − μ2 given by the interval are between −5 and 5, it is not likely that the difference will be more than 5 ppm, and hence the statistical significance of the difference is not of practical importance to the the engineers. Option C is correct.