As production manager in the new soft drink factory, you are worried that the machines don’t fill the bottles properly. Each bottle should be filled with soda, but random samples show that 48 soda bottles have an average of , with an empirical standard deviation of . You are wondering if you need to recalibrate the machines so that they become more precise.
This is a classic case of hypothesis testing by normal distribution. You now follow the instructions above and select % level of significance, since it is only a quantity of soda and not a case of life and death.
- The null hypothesis is that there is L in each bottle:
The alternative hypothesis in this case is that the bottles do not contain L and that the machines are not precise enough. This thus becomes a two-sided hypothesis test and you must therefore remember to multiply the -value by 2 before deciding whether the -value is in the critical region. This is because the normal distribution is symmetric, so . Thus it is just as likely to observe an equally extremely high value as an equally extreme low:
- Find the -value by calculating :
- You have chosen that if the machines are to be recalibrated your -value must be less than %. Therefore, there must be less than % chance of the machine filling L on average in the bottles. Your -value is
so that must be kept and the machines are fine as is.
Had the -value been less than the level of significance, that would have meant that the calibration represented by the alternative hypothesis is significantly better for the business.