hypothesis testing

Question

A washing machine manufacturer says that 90% of its washers last five years before repairs are necessary. A random sample of 100 washers showed that 79 of them lasted five years before they needed repair. Test to see if there is a difference between the manufacturer claim and the sample results. List the test statistic, the p-value, Does this data indicate that the manufacturer’s claim is either too high or too low? (yes or no)

Answer 1

The formula for the test statistic is:

 

 

                  ( pˆ−p0)

     z=    --------------------

               sqrt(p0(1−p0)/n)

 

 

Where p^ or the p-value is the sample proportion, which in this case is 79/100 = 79% = 0.79

 

p0 is the claimed proportion or the assumed population proportion per status quo,

         which in this case is 90% = 0.9

 

n = 100

 

The test statistic is z =  (0.79-0.9) / sqrt( 0.9 * (1-0.9) /100) 

 

                              = -(0.11)/ sqrt( 0.9 * 0.1 / 100)

 

                              = -0.11 / sqrt( 0.009)

 

                             = -0.11 / 0.03

 

                              =-3.666

 

This is well outside the range of the rejection region

for any confidence level up to 99.X%

 

There is strong statistical evidence that manufactures

claim is quite false. The proportion of washing machines

that have a lifespan of 5 years is LESS than 90%