Question

0 Let 3.1 be the true standard deviation of the normal population from which random sample is choosen, how large should the sample size be for testing H₂: H=5 against H: M = 5.5 in order that x=0001 and f = 0.05

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Answer 1

To determine the required sample size for testing H₂: H=5 against H: M = 5.5 with a significance level of 0.05 and a power of 0.001, we can use the following formula:

n = [((zα/2 + zβ)σ / δ)²]

where:

n is the required sample size
σ is the true standard deviation (given as 3.1)
δ is the difference between the two means to be tested (5.5 - 5 = 0.5)
zα/2 is the critical value for a two-tailed test at the 0.05 level of significance (1.96)
zβ is the critical value for a power of 0.001 (3.09)
Substituting the values, we get:

n = [((1.96 + 3.09) x 3.1 / 0.5)²]
n = [(5.05 x 3.1 / 0.5)²]
n = (31.155)²
n = 972.14

Rounding up to the nearest whole number, we get n = 973.

Therefore, the sample size should be at least 973 to test H₂: H=5 against H: M = 5.5 with a significance level of 0.05 and a power of 0.001.

Answer 2

To determine the sample size required to test the hypothesis, we can use the formula:

n = ((zα/2 + zβ) / (d/σ))^2

where:

n is the sample size

zα/2 is the critical value of the standard normal distribution at α/2 (the significance level divided by 2), which corresponds to the upper tail probability of 0.025 for a two-tailed test at a significance level of 0.05

zβ is the critical value of the standard normal distribution at β (the desired power), which corresponds to the lower tail probability for a two-tailed test

d is the difference between the hypothesized mean (5) and the true mean (5.5)

σ is the true standard deviation (3.1)

Substituting the given values, we get:

n = ((zα/2 + zβ) / (d/σ))^2

n = ((1.96 + zβ) / (0.5/3.1))^2

n = ((1.96 + zβ) / 0.16129)^2

n = (12.123 + 12.227zβ)^2

To solve for zβ, we need to specify the desired power of the test. Let's say we want a power of 0.8 (i.e., we want to have an 80% chance of rejecting the null hypothesis when it is false). The critical value for the standard normal distribution at a lower tail probability of 0.2 (corresponding to a power of 0.8) is approximately -0.84.

Substituting this value into the formula, we get:

n = (12.123 + 12.227(-0.84))^2

n = (12.123 - 10.285)^2

n = 3.838^2

n = 14.7 (rounded up to the nearest whole number)

Therefore, we would need a sample size of at least 15 to test the hypothesis with a significance level of 0.05, a true standard deviation of 3.1, a hypothesized mean of 5, and a true mean of 5.5, and achieve a power of 0.8