Question

A quiz is created by choosing for each question on the quiz 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 quiz 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 c∗ be the critical value for the rejection region on this question. Calculate E [ c∗ ].

Answer 1

ANSWER::

We assume that the significance level to be 0.05.

Critical value of z for 0.05 significance level is -1.645

Degree of freedom = n-1 = 35-1 = 34

Critical value of t for 0.05 significance leve and df = 34 is -1.691

For = 3

Standard error = 3 / = 0.5070926

Since we know the population standard deviation, we will use z score to estimate the critical value.

c* = 15 - 1.645 * 0.5070926

= 14.16583

For s = 4.2

Standard error = 4.2 / = 0.7099296

Since we do not know the population standard deviation, we will use t statistic to estimate the critical value.

c* = 15 - 1.691 * 0.7099296 = 13.79951

For s = 4.2

Standard error = 5.7 / = 0.9634759

Since we do not know the population standard deviation, we will use t statistic to estimate the critical value.

c* = 15 - 1.691 * 0.9634759 = 13.37076

The PMF of C* is,

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

= 13.77766

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