Answer to Question #214978 in Statistics and Probability for Madimetja

Question

For the production process of ballpoint pens of a certain type, it is known that 5% of the pens are defective. The manager believes that this percentage is too high and hence changes the production process. To test the merits of this new process, a random sample of 350 pens is taken; only 9 of them are defective.

a) (i) Determine a 90% confidence interval for p, the population proportion of defective pens produced with the new process.

ii) Interpret the confidence interval.

iii) Do you think that the new process is better?  

 

b) Conduct a two-sided test to conclude whether revising the process has indeed changed the population proportion of defective pens. Take a= 0:05.

i) State the null and alternative hypothesis.

ii) State and calculate the appropriate test statistic.

iii) Determine the critical value of the test and state the rejection region.

iv) State whether or not you reject the null hypothesis.

v) Draw an appropriate conclusion.

Answer 1

(a).

i. 90% confidence interval

$CI=\hat P\pm z_{\frac{\alpha}{2}}(\sqrt{\frac{\hat P(1-\hat P)}{n}})$

$z_{0.025}=1.96$

$\hat P=\frac{9}{350}$

$n=350$

$CI=\frac{9}{350}\pm1.96(\sqrt{\frac{\frac{9}{350}(1-\frac{9}{350})}{350}})$

$=\frac{9}{350}\pm0.01658$

$=(0.0091, 0.0423)$

$0.0091\le P\le0.0423)$

ii.

We are 95% confident that the population proportion of defective pens lie between 0.0091 and 0.0423.

iii.

The new process is better since it lowers proportion of defective pens significantly. Initial proportion of 0.05 is not within the 95% confidence interval.

(b)

i.

$H_0:P=0.05$

$H_a:P\ne0.05$

ii.

$np=\frac{9}{350}\times350=9>5$ and

$n(1-p)=\frac{341}{350}\times350=341>5$

Since both nP and n(1-P) are greater than 5, Z-score can be used. the test statistic is;

$z=\frac{\hat P-P}{\sqrt{\frac{P(1-P)}{n}}}$

$z=\frac{\frac{9}{350}-0.05}{\sqrt{\frac{0.05(1-0.05)}{350}}}$

$=-2.0847$

iii. critical value

$CV=z_{\frac{\alpha}{2}}$

$=z_{0.025}=\pm1.96$

iv.

Since the absolute value of the test statistic (2.0847) is greater than the absolute critical value (1.96), we reject the null hypothesis.

v

The new production process is better since there is a significant difference between the initial proportion of defectives and the new proportion.