A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 404.0404.0 gram setting. It is believed that the machine is underfilling the bags. A 4848 bag sample had a mean of 400.0400.0 grams. A level of significance of 0.020.02 will be used. Is there sufficient evidence to support the claim that the bags are underfilled

Bags are being underfilled by the machine

Step-by-step explanation:

Solution:-

- The potato chips manufacturer wants to statistically determine whether the machine is operating correctly based on its weight control.

- The weight control limit was placed on the machine for 404.0 grams.

- He believes that the machine is underfilled.

- So, he takes a sample of size n = 48 bags and weigh them separately and get the following data for his sample.

                 Sample Mean ( x^ ) = 400.0 grams

                 Variance ( σ ) = 169 grams

- He wants to statistically check his doubts on the machine operability at a significance level ( α ) of 0.02.

- We will make an hypothesis that the machine works fine. So,

                Null hypothesis: u = 404.0 grams

- We will test against what he believes to be the case i.e " under-filling of bags "

                Alternate hypothesis: u < 404.0 garms

- Next we will determine what test is to be applied in this case.

- For the above two conditions we can apply the use standard normal distribution ( Z-score ).

- We are testing for underfilling of bags. So we will apply one-left tail test.

- Hence, the critical value for the Alternate hypothesis to hold true would be:

                    P ( Z < Z-critical ) = α  

                    P ( Z < Z-critical ) = 0.02

                    Z-critical = -2.06  

- Next we compute the Z-statistics associated with the sample data obtained:

                    Z-test = ( x^ - u ) / √ ( σ^2 / n )

                    Z-test = ( 400 - 404 ) / √ ( 169 / 48 )

                    Z-test = -4 / 1.87638

                    Z-test = -2.13176

- For the left-one tailed test the Z-critical value provide the limit underneath which all the statistical test values lie in the rejection region.

Therefore,

                    -2.13176 < -2.06

                    Z-test < Z-critical.  ( Null hypothesis Rejected )

Conclusion:

- The statistics value lies in the rejection region; hence, the Null hypothesis is rejected. With can conclude with 98% confidence that the claim made on bags being underfilled by the machine is evident.

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