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 = 6.8 ppb arsenic, with s = 2.6 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 Student's t, since the sample size is large and σ is unknown. The standard normal, since the sample size is large and σ is known. 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 unknown. 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. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot

(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

Given that,
population mean(u)=8
sample mean, x =6.8
standard deviation, s =2.6
number (n)=31
null, Ho: μ=8
alternate, H1: μ<8
level of significance, α = 0.01
from standard normal table,left tailed t α/2 =2.457
since our test is left-tailed
reject Ho, if to < -2.457
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =6.8-8/(2.6/sqrt(31))
to =-2.5697
| to | =2.5697
critical value
the value of |t α| with n-1 = 30 d.f is 2.457
we got |to| =2.5697 & | t α | =2.457
make decision
hence value of | to | > | t α| and here we reject Ho
p-value :left tail - Ha : ( p < -2.5697 ) = 0.00769
hence value of p0.01 > 0.00769,here we reject Ho
ANSWERS
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a.
null, Ho: μ=8
alternate, H1: μ<8
b.
The Student's t, since the sample size is large and σ is unknown.
c.
test statistic: -2.5697
critical value: -2.457
decision: reject Ho
p-value: 0.00769
p value is less than alpha value.
d.
we have enough evidence to support the claim that the mean level of arsenic in this well is less than 8 ppb.
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