Question

Your company manufacturers electronic thermostats. Each thermostat uses a temperature sensitive switch to turn the heating/cooling system that it controls on or off. Periodically, you inspect a batch of switches from a large shipment of switches to verify quality of the work of your supplier. Let X = number of failures in the sample. Let p = the true proportion of failing switches in the shipment.

Suppose that in an SRS of 110 switches, you find 6 failures.

What is the upper endpoint of a 99% confidence interval for p, the population proportion of failing switches?

(You will need to calculate z* in Excel, and do not round in your intermediate calculations. Use the Wilson Estimate to calculate the interval.)

Express your answer in decimal form to two decimal places of accuracy.

Answer 1

Let X = number of failures in the sample.
Let p = the true proportion of failing switches in the shipment.

Suppose that in an SRS of 110 switches, you find 6 failures i.e. sample proportion (phat)=6/110

Upper endpoint of a 99% confidence interval for p, the population proportion of failing switches can be found out as follows:

Wilson estimate for interval is [phat-z*sqrt(phat*(1-phat)/n), phat+z*sqrt(phat*(1-phat)/n)].

Upper endpoint= phat + z*sqrt(phat*(1-phat)/n)

Computation in MS-EXCEL:

phat	6/110	0.054545455
1-phat	1-6/110	0.945454545
n		110
alpha		0.01
z		2.575829304	NORMSINV(1-0.01/2)
z*sqrt(phat*(1-phat)/n)		0.05577253
phat+z*sqrt(phat*(1-phat)/n)		0.110317985

Upper endpoint = 0.11