SPSS Confidence Intervals and Chi-Squared: Example Problems

Assume you are the manager of a mustard seed factory in Colombia. Your company has received complaints that there is not enough mustard seed in your economy size packages.
You ask your supervisor and chief operating officer Juan Valdez to test the new mustard seed packaging machine you are installing. He runs a sample of 36 packages, with the results of package sizes in ounces:

Dataset three
1.56 1.9 1.85 1.9 1.7 1.79
1.75 1.62 1.81 1.65 1.77 1.75
1.8 2.2 1.7 1.69 1.88 1.9
1.7 1.7 1.6 1.85 1.65 1.65
1.8 1.91 1.7 1.59 1.76 1.65
1.9 2.0 1.91 1.65 1.8 1.82

a. Calculate a 95% confidence interval (CI) on the average weight of packaged mustard seed. Explain very carefully to the packaging workers what the 95% confidence interval numbers mean. Include your SPSS output.

b. Now, let's assume that the package claims that it contains 1.7 oz. of mustard seed. Some of the customers claim that there isn't enough mustard seed in the 1.7 oz. economy size; corporate management is worried that there may be too much. Given the sample of 36, assuming that it is adequate, test the hypothesis that the average amount of mustard seed in the package meets the 1.7 oz. standard with, say, 95% confidence. Be sure to state the null and alternative hypothesis and which you support. Include your SPSS output.

c. Over a period of time, Juan determines that the true mean of the packages is in fact 1.72 ounces with standard deviation the same as the sample above. Assume that Juan has a strong consumer orientation. Juan is also a savvy businessman. Indeed, a local TV station has been checking up on Juan by periodically sampling 36 packages and figuring up the average weight. Juan tells you that he wants to be at least 95% sure that the average package has at least 1.7 ounces. Further, he wants to be sure that 95% of the time, the TV crew that is watching him will find a sample mean of at least 1.70 ounces. At the same time he doesn't want to set the equipment to fill packages with any more product than he absolutely has to. What advice would you give the mustard seed packaging factory management as to how to calibrate their equipment? In particular, should they increase or decrease the amount they are putting in the packages? Are there any other steps they might be able to take to improve the situation?

Chi-square

A market research firm conducts a brand awareness test on a new product. As marketing manager you suspect there is a difference between the awareness levels of males and females. To test this out you conduct a quick market research test of 100 respondents. You find:
Dataset 3 - 69 respondents - 35 men and 34 women are aware of the product. You also find that 31 respondents - 25 men and 6 women are not aware of the product.
What are your hypotheses (H0 and H1)? With 95% confidence, which do you support? What is the chi-square statistic? What is the probability value (labeled "significance" by SPSS)? Interpret this result. Be sure to include some of your SPSS output in your answer.

Hint: You may need to do a bit of keying here - perhaps 100 rows of data with two columns (gender and aware)?
Hint: In chi-square problems, H0: variables are independent, H1: variables are dependent
Source: Hanke, J.E. and Reitsch, A.G. (1994). Understanding Business Statistics, 2nd edition. Homewood, Ill.: Richard D. Irwin.