Practise modelling sample proportions by calculating centre and standard error, checking approximate normality, and interpreting standardised tail probabilities.
A large company produces an equal number of brand-name lightbulbs and generic lightbulbs. The director of
quality control sets guidelines that production will be stopped if there is evidence that the proportion of all
lightbulbs that are defective is greater than 0.10. The director also believes that the proportion of brand-name
lightbulbs that are defective is not equal to the proportion of generic lightbulbs that are defective. Therefore, the
director wants to estimate the average of the two proportions.
To estimate the proportion of brand-name lightbulbs that are defective, a simple random sample of
400 brand-name lightbulbs is taken and 44 are found to be defective. Let X represent the number of
brand-name lightbulbs that are defective in a sample of 400, and let pX represent the proportion of all
brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.
4 Complete Response
Three sections essentially correct
3 Substantial Response
Two sections essentially correct and one section partially correct
2 Developing Response
Two sections essentially correct and no sections partially correct
OR
One section essentially correct and one or two sections partially correct
OR
Three sections partially correct
1 Minimal Response
One section essentially correct
OR
No sections essentially correct and two sections partially correct
One condition for obtaining an interval estimate for pX is that the distribution of p^X is approximately
normal. Is it reasonable to assume that the condition is met? Justify your answer.
Part (a):
It is reasonable to assume that the distribution is approximately normal. The required condition is that
there are at least 10 successes and 10 failures in the sample. In this case there are 44 defective lightbulbs
and 356 non-defective lightbulbs, thus both exceed the minimum of 10 required.
How many standard errors is the observed value of p^X from 0.10 ?
To estimate the proportion of generic lightbulbs that are defective, a simple random sample of 400 generic
lightbulbs is taken and 104 are found to be defective. Let Y represent the number of generic lightbulbs that are
defective in a sample of 400. It is reasonable to assume that Y is a binomial random variable and the distribution
of p^Y is approximately normal, with an approximate standard error of 0.0219. It is also reasonable to assume
that X and Y are independent.
The parameter of interest for the manager of quality control is D, the average proportion of defective lightbulbs
for the brand-name and the generic lightbulbs. D is defined as D=2pX+pY.
Part (c):
Since p^X=0.11,0.11−0.10=0.01, so that p^X is 0.01560.01=0.64 standard error away from 0.10.
It was recently reported that in the United States, 40.3\% of all births are to unmarried mothers. A county health administrator is investigating whether births to unmarried women is higher in her county than in the national average. If so, she will propose additional funding to counsel unmarried mothers. A random sample of 100 births in the county will be looked at. Let p= the proportion of the population of women giving birth in the county who are unmarried. Consider the following hypotheses. The hypotheses are H0:p=0.403 and Ha:p>0.403.
For true p=0.403, calculate σp^.
σp^=(0.403)(0.597)/100=0.04905.
For true p=0.45, calculate σp^.
σp^=(0.45)(0.55)/100=0.04975.
Calculate the probability that the null hypothesis is rejected.
P(p^>0.4837)=P(Z>(0.4837−0.45)/0.04975)=0.249.