Question

16) Unfortunately, arsenic occurs naturally in some ground water (Reference: Union Carbide Technical Report K/UR-1). A mean arsenic level of ? = 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 37 tests gave a sample mean of ?̅ = 7.2 ppb arsenic, with ? = 1.9 ppb. Does this information indicate that the mean level of arsenic in this well is less than 8 ppb? Use ? = 0.01.

a) What is the level of significance? State the null and alternative hypotheses.

b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution of the sample test statistic.

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

d) Based on your answers in parts (a) through (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ??

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

Answer 1

## We have given :

n  = sample size  = 37

x̄ = sample mean  = 7.2

μ = population mean = 8

s = sample standard deviation  = 1.9

α = 1 % = 0.01

Claim  : Does this information indicate that the mean level of arsenic in this well is less than 8 ppb?

## a) What is the level of significance? State the null and alternative hypotheses.

Answer :

level of significance is 1 % ie 0.01 , and

## null and alternative Hypothesis :

Ho : μ = 7.2 vs H1: μ   < 7.2

( it is left tailed , one tailed test )

## b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution of the sample test statistic.

Answer : here sample follow normal distribution , and population variance is unknown hence we can use

one sample t test .

test statistics :   

t = (x̄ - μ) *√ n /  σ

t = ( 7.2  - 8 ) * √ 37 / 1.9

= - 4. 8662 / 1.9

= - 2.561163

## c) Find (or estimate) the P-value.

Answer : p value = P [ t <  - 2.561163 ]  

df = degree of freedom = n -1 = 36

use statistical table :

p value = 0.00738

## d) Based on your answers in parts (a) through (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ??

Answer :

Decision : we reject Ho if p value is less than  α value using p value approach here p value is less than  α value we reject Ho at given level of significance .

## e) Interpret your conclusion in the context of the application

Answer :

Conclusion : There is sufficient evidence to conclude that ,  this information indicate that the mean level of arsenic in this well is less than 8 ppb