A consumer products company is formulating a new shampoo and is interested in foam height (in mm). Foam height is approximately normally distributed and has a standard deviation of 20 mm. The company wishes to test H0: μ = 175 mm versus H1: μ > 175 mm, using a random sample of n = 10 samples. a) find the type I error probability α if the critical region is x bar >185.
b) what is the probability of type II error if the true mean foam height is 200 mm?
c) what is the power of the test from part (b)?

At 96% population = 2.05 z

/x = 185 x 0.04 =  7.4

/x = 185 = 177.6, 192.4

a) probability error is not giving the population mean.

or

a) they have used the random sample of 10 instead of 20mm

b) when using true mean foam height of 200m we find the 112.5% population change. So 112.5% = √200  = 14.1421356237^2

c) this shows us 11% change to find the mean = 50%  then 22% = 100%

22% would therefore be our maximum and show us √225 = 15^2 so would show 0.96 change when square rooted.

0.96^2 x 14.1421 =  0.9216 x 14.1421 = 13.03335936 = 1.97 which is our z factor.

One important property of confidence intervals (and standard errors) is that they vary inversely with the square root of the sample size. For example, if you were to quadruple your sample size, it would cut the SE in half, and it would cut the width of the CI in half. This “square root law” is one of the most widely applicable rules in all of statistics.

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