Question

Let x be a random variable that represents micrograms of lead per liter of water (µg/L). An industrial plant discharges water into a creek. The Environmental Protection Agency (EPA) has studied the discharged water and found x to have a normal distribution, with

σ = 0.7 µg/L.

† Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and thereby produce a slightly more "conservative" answer.

(a) The industrial plant says that the population mean value of x is

μ = 2.0 µg/L.

However, a random sample of

n = 10

water samples showed that

x = 2.53 µg/L.

Does this indicate that the lead concentration population mean is higher than the industrial plant claims? Use

a = 1%.

(i) What is the level of significance?

State the null and alternate hypotheses.

H₀: μ = 2.0; H₁: μ ≠ 2.0H₀: p = 2.0; H₁: p > 2.0    H₀: μ = 2.0; H₁: μ > 2.0H₀: p = 2.0; H₁: p < 2.0H₀: μ = 2.0; H₁: μ < 2.0H₀: p = 2.0; H₁: p ≠ 2.0

(ii) What sampling distribution will you use? What assumptions are you making?

The Student's t, since we assume that x has a normal distribution with unknown σ.The standard normal, since we assume that x has a normal distribution with unknown σ.    The Student's t, since we assume that x has a normal distribution with known σ.The standard normal, since we assume that x has a normal distribution with known σ.

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(iii) Find (or estimate) the P-value.

P-value > 0.5000.250 < P-value < 0.500    0.100 < P-value < 0.2500.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010

Sketch the sampling distribution and show the area corresponding to the P-value.  

(iv) Based on your answers in parts (i) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(v) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the population mean discharge level of lead is higher.There is insufficient evidence at the 0.01 level to conclude that the population mean discharge level of lead is higher.    

(b) Find a 95% confidence interval for μ using the sample data and the EPA value for σ. (Round your answers to two decimal places.)
lower limit     µg/Lupper limit     µg/L
(c) How large a sample should be taken to be 95% confident that the sample mean

x

is within a margin of error

E = 0.4 µg/L

of the population mean? (Round your answer up to the nearest whole number.)
water samples

Answer 1

As per the given question, data is as follows
Hypothesized mean = 2
Sample mean = 2.54
Standard deviation = 0.7
sample size, n = 10
(i) Significance level, α = 1% = 0.01

(ii)  Value of sample test statistics is given by

The null and the alternative hypotheses used for testing are given

(iii)

The test statistic is 2.57.

The p-value is 0.0051 can be obtained by entering the formula

“=(1-NORM.S.DIST(2.57,TRUE))”in EXCEL.

Decision rule:

If p-value ≤ α, then reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

The level of significance is 0.01

Conclusion:

The P-value is 0.0051 and the level of significance is 0.01.

Here, P-value is less than the level of significance.

Hence, the null hypothesis is rejected.

There is sufficient evidence to conclude that the lead concentration population mean is higher than the industrial plant.

(iv)

By using the results obtained in part (i) and (iii) the conclusion is made.

Decision Rule:

Reject the null hypothesis when the p-value is lesser than or equal to the level of significance. Otherwise, do not reject the null hypothesis.

Conclusion:

It is given that the p-value is lesser than 0.010 and the level of significance is 0.010.

The p-value is less than the level of significance.

Thus, the null hypothesis is rejected.

Hence, there is sufficient evidence to conclude that the lead concentration population mean is higher than the industrial plant claims at 1% level of significance.

The conclusion is that we reject the null hypothesis and conclude that the data are statistically significant