Question:

"The Geometric Distribution and Shorts in NiCad Batteries"

In their article "A Case Study of the Use of an Experimental Design

in Preventing Shorts in Nickel-Cadmium Cells"; Ophir, El-Gad, and

Snyder describe a series of experiments conducted in order to reduce

the proportion of cells being scrapped by a battery plant because of

internal shorts.

The experimental program was successful in reducing the proportion of

manufactured cells with internal shorts to around 0.03.

The following procedure is being developed in order to monitor the

process for increases in the true proportion of manufactured cells

with internal shorts.

Suppose that testing of batteries for internal shorts begins

on a production run in this plant for monitoring/control purposes,

let B = the number of batteries required until

the first internal short is discovered.

[a]

What is the distribution and expected value of B,

if the proportion of manufactured cells with internal shorts

remains at 0.03?

What is the distribution and expected value of B,

if the proportion of manufactured cells with internal shorts

increases to 0.10?

What happens, in general, to the expected value of B

as the proportion of manufactured cells with internal shorts

increases? Choose one: [Larger or Smaller]

[b]

Consider using B to test the following hypotheses:

H_0: % = 0.03 versus H_a: % > 0.03

where % = true proportion of manufactured cells with internal shorts.

B has been proposed as the TEST STATISTIC,

however three Rejection Regions are being considered.

A choice needs to be made concerning which of the three

Rejection Regions to implement in the procedure.

Rejection Region 1: Reject H_0 if B = 1.

Rejection Region 2: Reject H_0 if B <= 2.

Rejection Region 3: Reject H_0 if B <= 3.

[b.i]

Calculate the probability of a Type I Error for Rejection Region 1.

Calculate the probability of a Type I Error for Rejection Region 2.

Calculate the probability of a Type I Error for Rejection Region 3.

Based upon this information which Rejection Region do you recommend? WHY?

[b.ii]

Now suppose, in reality, the true proportion of manufactured cells

with internal shorts has increased to 0.10

Calculate the power for Rejection Region 1.

Calculate the power for Rejection Region 2.

Calculate the power for Rejection Region 3.

Based upon this information which Rejection Region do you recommend? WHY?

[b.iii]

For each Rejection Region determine an equation/formula/function that

relates power to %, true proportion of manufactured cells w/ internal shorts.

For Rejection Region 1: Power = ??function of %??

For Rejection Region 2: Power = ??function of %??

For Rejection Region 3: Power = ??function of %??

[b.iv]

Graph each of the above power functions on the same set of axes

use EXCEL, MATLAB, etc. to get a "nice" graph.

Place % on the x-axis and power on the y-axis.

Note: the range for both % and power is 0 to 1.

Now make a final recommendation concerning the Rejection Region

you would recommend. Justify your answer by giving clear reasoning

why you selected the Rejection Region you did.

[c]

Now suppose the experiment has been conducted and the observed value of B = 5.

Calculate the p-value.

My Answer so far:

(a) Given that, in batteries, an experimental process successfully reduced the proportion of manufactured cells with internal shorts to around 0.03. Let B = the number of batteries required until the first internal short is discovered. Given, the proportion of manufactured cells with internal shorts to around 0.03.

p = 0.03

q = 1 - p = 1 - 0.03 = 0.97

B ~ Geometric (p = 0.03)

P(X = x) = (0.03)*(0.97)^x for x = 1, 2,...

E(x) = q/p = 0.97/0.03 = 32.333

so, the expected value of B is 32

I'm wondering if my attempt at (a) is correct and if anyone has any helpful hints about how I should start part (b)