Question

Suppose that a bottled water plant wants to determine if the bacteria count in their water supply exceeds the safety threshold of 100 cfu/mL (colony-forming units per milliliter). Ten different samples were taken at random points in the water supply, and the bacterial count was measured in cfu/mL. The data is presented below:

104 102 100 97 93 105 102 97 96 102

Answer the following four questions on whether the bacteria count in the water supply exceeds the safety threshold of 100 cfu/mL.

Required:
a. Which is appropriate alternative hypothesis?
b. What is the value of t that defines the rejection region for a= 0.01?
c. Using the information in the above two questions, what is the value of the test statistic T?

Answer 1

a

The alternative hypothesis is
`H_a: \mu > 100 \ cfu/mL`

b

`z_(\alpha ) = -2.33`

c

`T = -1.7186`

d

the decision rule is

Fail to reject the null hypothesis

Explanation:

From the question we are told that

The data given is

104 102 100 97 93 105 102 97 96 102

The population mean is
`\mu = 100\ cfu/mL`

Generally the sample mean is mathematically represented as

`\= x = (\sum x_i)/(n)`

=>
`\= x = ( 104 + 102 + \cdots + 102)/(10)`

=>
`\= x = 99.8`

Generally the standard deviation is mathematically represented as

`\sigma = \sqrt{ (\sum (x_i - \= x)^2)/(n) }`

=>
`\sigma = \sqrt{ ( ( 104- 99.8)^2 + (102 - 99.8)^2 + \cdots + (102-99.8)^2)/(10) }`

=>
`\sigma = 3.68`

The null hypothesis is
`H_o : \mu = 100 \ cfu/mL`

The alternative hypothesis is
`H_a: \mu > 100 \ cfu/mL`

From the question we are told that the level of significance is
`\alpha = 0.01`

The critical value of
`\alpha` is obtained from the normal distribution table a the value is

`z_(\alpha ) = - 2.33`

Generally the test statistic is mathematically represented as

`T = ( \= x - \mu)/( (\sigma )/(√(n) ) )`

=>
`T = ( 99.8 - 100)/( (3.68 )/(√(10) ) )`

=>
`T = -1.7186`

From the value obtained and the value calculated we see that the critical value is not within the region of rejection(i.e -1.7186 to + -1.7186 ) hence we fail to reject the null hypothesis

Thus the decision rule is

Fail to reject the null hypothesis