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 = 2.54 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: μ = 2.0; H1: μ ≠ 2.0H0: p = 2.0; H1: p ≠ 2.0H0: p = 2.0; H1: p < 2.0 (ii) What sampling distribution will you use? What assumptions are you making? 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.The Student's t, since we assume that x has a normal distribution with unknown 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 fail to reject the null hypothesis and conclude the data are statistically significant.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. (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/L upper 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

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 = 2.54 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: μ = 2.0; H1: μ ≠ 2.0H0: p = 2.0; H1: p ≠ 2.0H0: p = 2.0; H1: p < 2.0 (ii) What sampling distribution will you use? What assumptions are you making? 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.The Student's t, since we assume that x has a normal distribution with unknown 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 fail to reject the null hypothesis and conclude the data are statistically significant.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. (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/L upper 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

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100%

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 = 2.54 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: μ = 2.0; H₁: μ ≠ 2.0H₀: p = 2.0; H₁: p ≠ 2.0H₀: p = 2.0; H₁: p < 2.0

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

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.The Student's t, since we assume that x has a normal distribution with unknown 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 fail to reject the null hypothesis and conclude the data are statistically significant.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.

(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/L
upper limit	µg/L

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

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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