Question

Q9-
According to data from the Environmental Protection Agency, the average daily water consumption for a household of four people in the United States is at least 243 gallons. Suppose a state agency plans to test this claim using an alpha level equal to 0.05 and a random sample of 100 households with four people. Assume that the population standard deviation is known to be 40 gallons. Calculate the probability of committing a Type II error if the true population mean is 230 gallons.

Select one:

a. 0.0331

b. 0.0712

c. 0.1412

d. 0.0537

Answer 1

The values of sample mean X̅ for which null hypothesis is rejected
Z = ( X̅ - µ ) / ( σ / √(n))
Critical value Z(α/2) = Z( 0.05 /2 ) = ± 1.645
1.645 = ( X̅ - 243 ) / ( 40 / √( 100 ))
X̅ <= 236.42
P ( X̅ > 236.42 | µ = 230 ) = 0.0537
P ( Type II error ) ß = 0.0537

Answer 2

To calculate the probability of committing a Type II error, we need to determine the critical value for the alternative hypothesis and calculate the corresponding probability.

Given: Population mean (μ) = 230 gallons Sample size (n) = 100 households Population standard deviation (σ) = 40 gallons Alpha level (α) = 0.05 (significance level)

First, let's calculate the critical value for the alternative hypothesis. Since the alternative hypothesis is that the true population mean is less than 243 gallons, we will use a one-tailed test with the alpha level of 0.05.

To find the critical value, we can use the z-score formula:

z = (x - μ) / (σ / √n)

where x is the hypothesized mean (243 gallons), μ is the population mean (230 gallons), σ is the population standard deviation (40 gallons), and n is the sample size (100 households).

z = (243 - 230) / (40 / √100) z = 13 / 4 z = 3.25

Next, we need to find the probability associated with this z-score using a standard normal distribution table or a statistical software. In this case, we are interested in the probability of getting a z-score greater than or equal to 3.25.

Using a standard normal distribution table or a statistical software, the probability of getting a z-score greater than or equal to 3.25 is approximately 0.00135.

Therefore, the probability of committing a Type II error is 0.00135.

The closest option among the given choices is a) 0.0331.