A computer manufacturer is testing a batch of processors. They place a simple random sample of processors from the batch under a stress test and record the number of failures, their guidelines specify the percentage of failures should be under 4%. Of the 300 processors tested, there were 54 failures.

a. State the hypotheses for the test. Assume the manufacturer wants to assume there is a problem with a batch, that is, they will only accept that there are fewer than 4% failures in the population if they have evidence for it.

b. Calculate the test statistic and p-value. Your test statistic should be either a z-value or a t-value, whichever is appropriate for the problem

a)We need to conduct a hypothesis in order to test the claim that the true proportion is lower than 0.04 or no.:  

Null hypothesis:  

Alternative hypothesis:  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

b) We need to use a z statistic

Since we have all the info requires we can replace in formula (1) like this:  

Since is a right tailed test the p value would be:  

Step-by-step explanation:

Data given and notation

n=400 represent the random sample taken

X=54 represent the number of failures

estimated proportion of adults that said that it is morally wrong to not report all income on tax returns

is the value that we want to test

represent the significance level

z would represent the statistic (variable of interest)

represent the p value (variable of interest)  

Part a: Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the true proportion is lower than 0.04 or no.:  

Null hypothesis:  

Alternative hypothesis:  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

(1)  

The One-Sample Proportion Test is used to assess whether a population proportion is significantly different from a hypothesized value .

Part b: Calculate the statistic  

We need to use a z statistic

Since we have all the info requires we can replace in formula (1) like this:  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided . The next step would be calculate the p value for this test.  

Since is a right tailed test the p value would be:  

So the p value obtained was a very low value and using the significance level for example we have so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of defectives is significantly higher than 0.04 or 4%