Consider the hypothesis h0 0 ha 0 where 2 is known it is...

Consider the hypothesis: H0 :μ=μ0 Ha :μ̸=μ0 where σ2 is known. It is known that for detecting a true mean of μ1 ̸= μ0 with probability 1 − β and type I error α, the required sample size is approximately: n ≈ [(Zα/2 + Zβ )σ/δ]^2 (a) Let us consider that μ0 = 15 and σ2 = 9. If the true mean is 20, what sample size must be used to ensure that the probability of type II error is no greater than 0.10? Assume that α = 0.05. (b) Derive the expression for n in 7(a).

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00:01 Okay, so i see that you need help with these questions, and it says, consider a random sample size of 20 from a normal distribution where the population mean is 11 and the standard deviation, i'm sorry, the sample mean is 11 and the standard deviation is 16.

00:23 Okay.

00:24 Assume that we know that the population standard deviation is four and it wants you to test whether it's greater than 12 or less than 12 at a size of 0 .01.

00:42 One.

00:43 So first thing that you have to do is you have to determine what the test statistic and looking at the t score, it's going to be approximately two point or looking at your your t -value chart it's going to be approximately if you take 11 minus 12 and you divide you multiply that by 2 divided by the square root of 20 and because the sample size is going to be 20 because you're taking your population variance.

01:38 Sorry, it's giving you your population variance.

01:40 So this is, let me just label this.

01:43 This is your population variance, which means your population standard deviation is four.

01:57 And this is your sample standard, i'm sorry, sample variance, which means that your sample standard deviation is 4 okay and the sample size is 20 so that's 20 and so if you do the 11 so this is for a 11 minus 12 and then 2 divided by the square root root of 20, that is going to be negative 1 times 2 to the square root of 20.

02:43 That is going to be, and i'll just use my calculator, negative 2 .236.

02:57 Determining the critical value to make that decision, well, the alpha is 0 .01, and the critical value from the standard normal table is negative 2 .33 so since the calculated z score is or the z score is greater than negative 2 .33 three, we fail to reject the null hypothesis.

03:39 Okay.

03:41 Then part b, it says, what is the beta equals p type two error if in fact the population mean is 10 .25? okay...

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