A popcorn company uses two machines to add butter to their microwave popcorn bags. The machine making the “movie theater style” bags is supposed to add \(5\)g of butter and has a standard deviation of \(1.2\)g. The machine making the “lightly buttered” option is supposed to add \(4\)g of butter with a standard deviation of \(1\)g. A quality control employee at the company wants to test if the machines are calibrated correctly. They decide to do a simple random sample of \(20\) bags from each machine and obtain sample means \(4.7\) and \(4.3\) from the movie theater style and lightly buttered style respectively. So, they set up a hyothesis test:

\(H_0 : \mu_1 - \mu_2 = 1\)

\(H_A : \mu_1 - \mu_2 \neq 1\)

  1. Based on the hypotheses, is \(\mu_1\) meant to represent the “movie theater style” or the “lightly buttered” style option?

  2. Will your model for the sampling distribution be the normal distribution or the \(t\)-distribution and why?

  3. Draw a picture of the sampling distribution. (you don’t need to include the picture here, just make sure to draw one)

  4. Test if the data collected provides statistically significant evidence that the amount of butter in the two styles is different than \(1\)g at a \(10 \%\) significance level.

  5. Now suppose that \(20\) bags from each group are sampled but we have not yet computed what their sample means are. Fill in the blank for the following statements using a significance level of \(\alpha=.1\):

  6. To compare their results, the quality control agent now wants to form a \(90 \%\) confidence interval for the true difference. Find this interval; does this interval support the claim that the true difference is \(1\)g? Why or why not?

  • “If the mean amount of butter in the movie theater style sample is less than _____ grams more than the lightly buttered sample, the data is statistically significant evidence that the difference is different than 1g.” (this is finding the cut-off values or critical values)

  • “If the mean amount of butter in the movie theater style sample is more than _____ grams more than the lightly buttered sample, the data is statistically significant evidence that the difference is different than 1g.” (this is finding the cut-off values or critical values)

Sleep deprived transportation workers. The National Sleep Foundation conducted a survey on the sleep habits of randomly sampled transportation workers and a control sample of non-transportation workers. The results of the survey are shown below.

hrs of sleep Control Pilots
<6 35 19
6-8 193 132
>8 64 51
Total 292 202
  1. If we want to test for statistically significant evidence of a difference between the proportions of truck drivers and non-transportation workers who get less than 6 hours of sleep per day what should our hypotheses be.

$H_0 : $

$H_A : $

  1. Will your model for the sampling distribution be the normal distribution or the \(t\)-distribution and why?

  2. Draw a picture of the sampling distribution. (you don’t need to include the picture here, just make sure to draw one)

  3. At the \(5 \%\) significance level, is there statistically significant evidence of a difference?

  4. Compare this to a \(95\%\) confidence interval for the difference in proportions (we have never done an example quite like this one before).