Question

A Statistics exam is created by choosing for each question on the exam one possible version at random from a bank of possible versions of the question. There are 20 versions in the bank for each question. A specific question on the exam involves a one-sample test for the population mean with hypotheses

H0 : µ = 15
Ha : µ < 15

with all versions of the question involving a sample of size n = 35.

Seven versions of the question give the population standard deviation as σ = 3. Six versions give the sample standard deviation as s = 4.2. The remaining versions give the sample standard deviation as s = 5.7. Let σ be the critical value for the rejection region on this question. Calculate E[c].

Answer 1

Answer: E (c *) = 13.77766

Explanation:

the significance level of 0.05 is known

seeks to calculate the sample for the population medis using the hypothesis

H0: µ = 15

H0: µ 15

sample size n = 35

Degrees of freedom df = n-1 = 35-1 = 34

The critical value for z is -1.645

The critical value for t is -1.691

For \ sigma = 3,

Standard error = 3 /√ 35 = 0.5070926

When obtaining the results of the population standard deviation, we will use the z-score to estimate the critical value.

ex = 15-1645 * 0.5070926 = 14.16583

For s = 4.2,

Standard error = 4.2 / √ 35 = 0.7099296

When we do not know what the population standard deviation is, we can use the statistical t to obtain the critical value

c * = 15-1.691 * 0.7099296 = 13.79951

for s = 5.7,

Standard error = 5.7 / √ 3.5 = 0.9634759

c * = 15-1.691 * 0.9634759 = 13.37076

It is obtained that c * is the critical value for the rejection region in this question

P (c * = 14.16583) = 7/20

P (c * = 13.79951) = 6/20

P (c * = 13.37076) = 1-7 / 20-6 / 20

= 7/20

E (c *) = (7/20) * 14,16583 + (6/20) * 13.79951 + (7/20) * 13.37076

Outcome:

E (c *) = 13.77766