Sparr Investments, Inc., specializes in tax-deferred investment opportunities for its clients. Recently Sparr offered a payroll...

Sparr Investments, Inc., specializes in tax-deferred investment opportunities for its clients. Recently Sparr offered a payroll deduction investment program for the employees of a particular company. Sparr estimates that the employees are currently averaging $100 or less per month in tax-deferred investments. A sample of 35 employees will be used to test Sparr's hypothesis about the current level of investment activity among the population of employees. Assume the employee monthly tax-deferred investment amounts have a population standard deviation of $75 and that a 0.05 level of significance will be used in the hypothesis test.

(a)

What is the probability of the type II error if the actual mean employee monthly investment is $125? (Round your answer to four decimal places. If it is not possible to commit a type II error enter NOT POSSIBLE.)

(b)

What is the probability of the type II error if the actual mean employee monthly investment is $130? (Round your answer to four decimal places. If it is not possible to commit a type II error enter NOT POSSIBLE.

(c)

Assume a sample size of 70 employees is used and repeat parts (b) and (c).

Solutions

Expert Solution

(a)

3 steps...

(i) Find the critical value - the value that separates the rejection region from fail to reject region.

= Ho value + z * sigma/sqrt(n)
= 100 + 1.645 * 75 / sqrt(35)
= 120.8542

(ii) Find the z-score for the xbar critical value using the true value for mu

z = (-)/(/sqrt(n))
z = (120.8542-125)/(75/sqrt(35))
z = -0.327

(iii) Find the probability that z < -0.327 (since we are testing greater than)

beta = P(xbar crit < 120.8542 given = 125)
beta = P(z < -0.327)
beta = 0.0005

the probability of the type II error if the actual mean employee monthly investment is $125 is 0.0005

(b)

xbar = 100 + 1.645 * 75 / sqrt(35) = 120.8542
z = (120.8542-130)/(75/sqrt(35)) = -0.7214
beta = 0.2358

the probability of the type II error if the actual mean employee monthly investment is $130 is 0.2358

(c)

* When mu = 125 and n = 70

= 100 + 1.645 * 75 / sqrt(70) = 113.763

z = (113.763-125)/(75/sqrt(70)) = -1.343

beta = 0.3669

* When mu = 130 and n = 70

= 100 + 1.645 * 75 / sqrt(70) = 113.79

z = (113.763-130)/(75/sqrt(70)) = -1.94

beta = 0.0262

orchestra answered 1 year ago

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