Question

For each of the following: a) state the null and alternative hypotheses

                                                    b) state the type I & type II errors

                                                    c) state the consequences of each error

                                                    d) is one more serious? Explain.

1) You have created a new manufacturing method for producing widgets which you claim will reduce the time necessary for assembling the parts. Currently it takes 75 seconds to produce a widget. The retooling of the plant for this change is very expensive and will involve a lot of down-time.

2) You are interested in building a new restaurant in an area if the income level is high enough to support the business. In order to open the restaurant, you estimate that a per capita income of $45,000 is needed.

3) A manufacturer of handheld calculators receives very large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is selected for inspection. The shipment is rejected only if the proportion of defective circuits is more than 5% and the entire shipment is returned to the supplier due to inferior quality.

Answer 1

Soln

1)

a)

Null and Alternate Hypothesis

H₀: µ = 75 (Time taken to produce a widget is 75)

H_a: µ < 75 (Time taken to produce a widget is less than 75)

b)

Type I Error

Type I error is the rejection of a true null hypothesis.

Type II Error

Type II error is the non-rejection of a false null hypothesis.

c)

Type I Error

We concluded that the Time taken to produce a widget is less than 75 when in fact Time taken to produce a widget is 75

Type II Error

We concluded that the Time taken to produce a widget is 75 when in fact Time taken to produce a widget is less than 75.

d)

Type I Error is more serious as it might result in opportunity loss (Revenue loss) since the retooling of the plant for this change is very expensive and will involve a lot of down-time due to our incorrect conclusion that Time taken to produce a widget is less than 75

2)

a)

Null and Alternate Hypothesis

H₀: µ = 45000 (Per Capita income is $45000)

H_a: µ > 45000 (Per Capita income is greater than $45000)

b)

Type I Error

Type I error is the rejection of a true null hypothesis.

Type II Error

Type II error is the non-rejection of a false null hypothesis.

c)

Type I Error

We concluded that Per Capita income is greater than $45000 when in fact Per Capita income is $45000

Type II Error

Per Capita income is greater than $45000 when in fact Per Capita income is $45000

d)

Type I Error is more serious as it might result revenue loss in new restaurant business as the Income level is not high enough to support the business

3)

a)

Null and Alternate Hypothesis

H₀: p = 5% (Proportion of defective circuit is 5%)

H_a: µ > 5% (Proportion of defective circuit is more than 5%)

b)

Type I Error

Type I error is the rejection of a true null hypothesis.

Type II Error

Type II error is the non-rejection of a false null hypothesis.

c)

Type I Error

We concluded that Proportion of defective circuit is more than 5% when in fact Proportion of defective circuit is 5%

Type II Error

We concluded that Proportion of defective circuit is 5% when in fact Proportion of defective circuit is more than 5%

d)

Type II Error is more serious as it might result in opportunity loss (Client/Revenue loss) since we concluded that Proportion of defective circuit is 5% when in fact Proportion of defective circuit is more than 5%. Due to our incorrect conclusion, client might go to another manufacturer for future orders