Question

Unfortunately, arsenic occurs naturally in some ground water†. A mean arsenic level of μ = 8.0 parts per billion (ppb) is considered safe for agricultural use. A well in Texas is used to water cotton crops. This well is tested on a regular basis for arsenic. A random sample of 31 tests gave a sample mean of x = 6.7 ppb arsenic, with s = 2.7 ppb. Does this information indicate that the mean level of arsenic in this well is less than 8 ppb? Use α = 0.01.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ > 8 ppb; H₁: μ = 8 ppb H₀: μ = 8 ppb; H₁: μ ≠ 8 ppb     H₀: μ = 8 ppb; H₁: μ < 8 ppb H₀: μ < 8 ppb; H₁: μ = 8 ppb H₀: μ = 8 ppb; H₁: μ > 8 ppb

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The standard normal, since the sample size is large and σ is unknown. The Student's t, since the sample size is large and σ is unknown.     The Student's t, since the sample size is large and σ is known. The standard normal, since the sample size is large and σ is known.

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


(c) Estimate the P-value.

P-value > 0.250 0.100 < P-value < 0.250     0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010

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

(d) Based on your answers in parts (a) to (c), 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 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. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

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

There is sufficient evidence at the 0.01 level to conclude that the mean level of arsenic in the well is less than 8 ppb.

There is insufficient evidence at the 0.01 level to conclude that the mean level of arsenic in the well is less than 8 ppb.  

Answer 1

Unfortunately, arsenic occurs naturally in some ground water†. A mean arsenic level of μ = 8.0 parts per billion (ppb) is considered safe for agricultural use. A well in Texas is used to water cotton crops. This well is tested on a regular basis for arsenic. A random sample of 31 tests gave a sample mean of x = 6.7 ppb arsenic, with s = 2.7 ppb. Does this information indicate that the mean level of arsenic in this well is less than 8 ppb? Use α = 0.01.

(a) What is the level of significance?

Level of significance = 0.01
The null and alternate hypotheses.

H0: μ = 8 ppb; H1: μ < 8 ppb

(b) The sampling distribution we use is,

The Student's t, since the sample size is large and σ is unknown.   

Standard error = s / sqrt(n) = 2.7 / sqrt(31) = 0.4849343

Test statistic, t = () / Std Error = (6.7 - 8) / 0.4849343 = -2.681

(c)

Degree of freedom = n-1 = 31-1 = 30

P-value = P(t < -2.681) = 0.0059

P-value < 0.010

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

(d)

Since p-value < α

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(e)

There is sufficient evidence at the 0.01 level to conclude that the mean level of arsenic in the well is less than 8 ppb.