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Hypothesis Testing, Significance Levels, Critical Values and Decision Rule

Please see the attached file for the fully formatted problems.

Faced with rising fax costs, a firm issued a guideline that transmissions of 10 pages or more should be sent by 2-day mail instead. Exceptions are allowed, but they want the average to be 10 or below. The firm examined 35 randomly chosen fax transmissions during the next year, yielding a sample mean of 14.44 with a standard deviation of 4.45 pages.

(a) At the .01 level of significance, is the true mean greater than 10?

One-tailed test.

Hypotheses:
H0:
H1:

Decision Rule:
Right-tail test.

Reject H0 if z > 2.326
Otherwise do not reject H0

Calculate the Statistic:

Conclusion:
Reject the null hypothesis.

No, the true mean is not greater than 10.

(b) Use Excel to find the right-tail p-value.

p-value = 1

A coin was flipped 60 times and came up heads 38 times.

(a) At the .10 level of significance, is the coin biased toward heads? Show your decision rule and calculations.

Hypotheses:
H0:
H1:

Decision Rule:
Right-tail test.

Reject H0 if z > 1.283
Otherwise do not reject H0

Calculate the Statistic:

This is where I get stuck - how do I complete any sort of calculation without the standard deviation??

(b) Calculate a p-value and interpret it.

The Web-based company Oh Baby! Gifts has a goal of processing 95 percent of its orders on the same day they are received. If 485 out of the next 500 orders are processed on the same day, would this prove that they are exceeding their goal, using α = .025?

Hypotheses:
H0: p>0.95
H1:

Check for normal distribution:
np>5 500*0.95 = 475 > 5
n(1-p)>5 500(1-0.95) = 25 > 5
Normal distribution

Test Statistic - Z-test:

Critical Value:

2 is greater than the critical value 1.960, so you reject the null statement.

They are not exceeding their goal at the .025 significance level.

An auditor reviewed 25 oral surgery insurance claims from a particular surgical office, determining that the mean out-of-pocket patient billing above the reimbursed amount was $275.66 with a standard deviation of $78.11.

(a) At the 5 percent level of significance, does this sample prove a violation of the guideline that the average patient should pay no more than $250 out-of-pocket? State your hypotheses and decision rule.

Hypotheses:
H0:
H1:

Decision Rule:
Right-tail test.

Reject H0 if z > 1.645
Otherwise do not reject H0

Calculate the Statistic:

Conclusion:
1.645 > 1.643 which would mean you would not reject the null statement. This would mean that at a 5 percent level of significance, patients are paying less than or equal to $250 out-of-pocket.

(b) Is this a close decision?

Extremely close - a few more patients that paid heavily could have skewed this tremendously and you would have more patients paying too much out-of-pocket than you are supposed to.

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