Question

Rachel measured the lengths of a random sample of 100 screws. The mean length was 2.9 inches, and the population standard deviation is 0.1 inch. To see if the batch of screws has a significantly different mean length from 3 inches, what would the value of the z-test statistic be?

Answer 1

-10

Explanation:

If we first note the denominator of fraction numerator sigma over denominator square root of n end fraction equals fraction numerator 0.1 over denominator square root of 100 end fraction equals fraction numerator begin display style 0.1 end style over denominator 10 end fraction equals 0.01

Then, getting the z-score we can note it is z equals fraction numerator x with bar on top minus mu over denominator begin display style 0.01 end style end fraction equals fraction numerator 2.9 minus 3 over denominator 0.01 end fraction equals negative 10

This tells us that 2.9 is 10 standard deviations below the value of 3, which is extremely far away.

Answer 2

z = 10

Explanation:

The value of the z-statistic is given by:

`z = (X - \mu)/(s)`

In which:

X is the measured value.

`\mu` is the expected value.

`s = (\sigma)/(√(n))` is the standard deviation of the sample.
`\sigma` is the standard deviation of the population.

In this question:

The mean length was 2.9 inches, and the population standard deviation is 0.1 inch.

This means that
`\mu = 2.9, \sigma = 0.1`

Random sample of 100 screws.

This means that n = 100.

To see if the batch of screws has a significantly different mean length from 3 inches, what would the value of the z-test statistic be?

3 inches, so
`X = 3`

`s = (0.1)/(√(100)) = 0.01`

`z = (X - \mu)/(s)`

`z = (3 - 2.9)/(0.01)`

`z = 10`