Question

Let μ denote the true average tread life of a certain type of tire. consider testing h0: μ = 30,000 versus ha: μ > 30,000 based on a sample of size n = 16 from a normal population distribution with σ = 1500. a test with α = 0.01 requires zα = z0.01 = 2.33. the probability of making a type ii error when μ = 31,000 is

Answer 1

To find the probability of making a type II error when μ = 31,000, we can use the power of the test. The power of a test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. In this case, the power of the test is approximately 1, which means that the probability of making a type II error is very low.

Step-by-step explanation:

To find the probability of making a type II error when μ = 31,000, we need to calculate the power of the test. The power of a test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.

Since we are testing whether μ > 30,000, the alternative hypothesis is μ > 30,000.

The power of the test can be calculated using the formula:

power = 1 - P(rejecting H0 | H1 is true)

Where P(rejecting H0 | H1 is true) is the probability of rejecting the null hypothesis when the alternative hypothesis is true.

Since the test statistic follows a normal distribution, we can calculate the power of the test using the formula:

power = 1 - P(z < zα + (μ - μ0)/(σ/√n))

Where zα is the z-score corresponding to the significance level α, μ is the true population mean, μ0 is the null hypothesis mean, σ is the population standard deviation, and n is the sample size.

In this case, zα = 2.33, μ = 31,000, μ0 = 30,000, σ = 1500, and n = 16. Plugging these values into the formula, we get:

power = 1 - P(z < 2.33 + (31,000 - 30,000)/(1500/√16))

power = 1 - P(z < 2.33 + 2.67)

power = 1 - P(z < 5)

Using a standard normal table, we find that P(z < 5) is very close to 1.

Therefore, the power of the test is approximately 1, which means that the probability of making a type II error when μ = 31,000 is very low.

Answer 2

The probability of making a type II error is given by one minus the power of the hypothesis test.

In general for the alternative hypothesis ,
`H_a:\mu\ \textgreater \ \mu_0`

The power of a hypothesis test is given by:


`\beta(\mu')=\phi\left(X \ \textless \ \mu_0+z_(1-\alpha)(\sigma)/(√(n))|\mu'\right) \\ \\ =\phi\left(z_(1-\alpha)+(\mu_0-\mu')/(\sigma/√(n))\right) =\phi\left(z_(1-0.01)+(30000-31000)/(1500/√(16))\right) \\ \\ =\phi\left(z_(0.99)+(-1000)/(1500/4)\right)=\phi\left(2.33-(1000)/(375)\right)=\phi(2.33-2.667) \\ \\\phi(-0.33)=0.3682`

The probability of making a type ii error when μ = 31,000 is given by


`1-\beta(\mu')=1-0.3682 \\ \\ =0.6318`