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 P-value if the sample average is = 185? Round your final answer to 3 decimal places.

(b) What is the probability of type II error if the true mean foam height is 200 mm and we assume that α = 0.05? Round your intermediate answer to 1 decimal place. Round the final answer to 4 decimal places.

(c) What is the power of the test from part (b)? Round your final answer to 4 decimal places.

Part a

The test statistic and P-value of the test is 1.581 and 0.057.

Part b

The type-II error is 0.0088.

Part c

The power of the test is 0.9912.

Step-by-step explanation:

The Z− test is a statistical test for the mean of a population. It can be used when or when the population is normally distributed and population standard deviation is known.

The type-II error is defined as the probability of accepting the null hypothesis when it is actually false. It is denoted by the Greek letter beta .

The power of a test against a specific alternative is the probability that the test will reject at a chosen significance level when the specified alternative value of the parameter is true.

The formula for the Z− test is,

Here, represents the sample mean

represents the population mean

represents the population standard deviation

represents the sample size

The formula for the type-II error is,

The formula for the power is,

Here, represents the type-II error

a)

The sample size is

The population standard deviation is,

The sample average is

Null hypothesis

Calculate the test statistic value.

Calculate the P− value of the hypothesis test.

b)

Calculate the type-II error.

c)

Calculate the power of the test.

x= 149. 24+7=31. 180-31=149

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step-by-step explanation: