Question

A research study investigated the adhesion of various biofilms to solid surfaces for possible use in environmental technologies. Suppose that for the bacterial strain Acinetobacter, five measurements gave readings of 2.69, 5.76, 2.67, 1.62, and 4.12 dyne-cm2. Assume that the standard deviation is known to be 0.66 dyne-cm2 Find a 95% two-sided confidence interval for the mean adhesion.

Answer 1

`3.372-1.96(0.66)/(√(5))=2.793`

`3.372+1.96(0.66)/(√(5))=3.951`

We are 95% confident that the true mean for the adhesion to solid surfaces in dyne-cm2 is between (2.793; 3.951)

Explanation:

Data provided

2.69, 5.76, 2.67, 1.62, and 4.12

We can calculate the sample mean with this formula:

`\bar X =(\sum_(i=1)^n X_i)/(n)`

And replacing we got:

`\bar X=3.372` represent the sample mean

`\mu` population mean (variable of interest)

`\sigma=0.66` represent the population standard deviation

n=5 represent the sample size

Confidence interval :

The two sided confidence interval for the true mean is given by:

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

We have the confidence level given of 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that
`z_(\alpha/2)=1.96`

Replacing into the formula for the interval we have this:

`3.372-1.96(0.66)/(√(5))=2.793`

`3.372+1.96(0.66)/(√(5))=3.951`

We are 95% confident that the true mean for the adhesion to solid surfaces in dyne-cm2 is between (2.793; 3.951)