You have a SRS of size n = 16 from a normal distribution with σ = 1.You wish to test H₀: μ = 0 versus Ha: μ 0.You decide to reject the null hypothesis if 0 and not reject the null hypothesis otherwise.Find the probability of a type II error when μ = 0.2.
A) 0.2119
B) 0.4801
C) 0.5199
D) Cannot be determined without additional information.

Question

You have a SRS of size n = 16 from a normal distribution with σ = 1.You wish to test H₀: μ = 0 versus Ha: μ > 0.You decide to reject the null hypothesis if  > 0 and not reject the null hypothesis otherwise.Find the probability of a type II error when μ = 0.2.
A) 0.2119 
B) 0.4801 
C) 0.5199 
D) Cannot be determined without additional information.

Quizplus · Accepted Answer

Since we know that the population standard deviation (σ) is 1 and the null hypothesis is μ = 0, we can calculate the standardized test statistic under the alternative hypothesis as:

z = (x̄ - μ) / (σ / √n)

where x̄ is the sample mean and n = 16.

When μ = 0.2, we can calculate the probability of a type II error (β) as follows:

β = P(z ≤ z_crit) = P(z ≤ z_crit | μ = 0.2)

where z_crit is the critical value that satisfies the condition 11eb86f3_5f65_4805_883a_5fdeb632a0bf_TB8195_11 > 0.

To find z_crit, we need to solve for z in the equation:

11eb86f3_5f65_4805_883a_5fdeb632a0bf_TB8195_12 = (z_crit - z) / σ/√n

Plugging in n = 16, σ = 1, and μ = 0.2, we get:

11eb86f3_5f65_4805_883a_5fdeb632a0bf_TB8195_13 = (z_crit - 0.2) / 0.25

Solving for z_crit, we get:

z_crit = 0.2 + 0.25(11eb86f3_5f65_4805_883a_5fdeb632a0bf_TB8195_13) = 0.2618

Using a standard normal table or calculator, we can find:

β = P(z ≤ 0.2618 | μ = 0.2) = 0.4801

Therefore, the probability of a type II error when μ = 0.2 is 0.4801.

You Have a SRS of Size N = 16 from a Normal