Previous studies have shown that the antibody titers in obese people are lower after vaccination than in people of normal weight. One suggested reason is that the vaccines may not effectively penetrate the layer of subcutaneous fat in obese individuals. To test this, Middleman et al. (2010) compared the response to hepatitis B virus vaccine in obese participants in two different groups. The researchers vaccinated one group of 10 individuals with standard 1-inch (2.5 cm) needles. They used 1.5-inch (3.8 cm) needles instead for a second group of 14 individuals. They later measured the antibody titers (in units of mIU/ml) of each participant. Greater numbers indicate a more successful response to the vaccine. These results are as follows.

a. What is the most-plausible range of values for the difference in mean antibody titers between the long- and short-needle groups? Use the 95% confidence interval to answer this question.

b. Use an appropriate hypothesis test to compare the means of the two groups. What can you conclude about the effectiveness of the vaccine as a function of the length of the needle?

c. What is the 95% confidence interval for antibody titer in the longneedle group?

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Short-needle group: 51.6,51.6, 87.4.87.4, 143.6,143.6, 144.6,144.6, 189.7,189.7, 189.8.189.8, 208.9.208.9, 324.7,324.7, 368,368, 383.9383.9 Long-needle group: 28.0,28.0, 181.6,181.6, 203.9.203.9, 243, 249.6, 249.6, 274.3.274.3, 341.2.341.2, 349.6,349.6, 393.0.393.0, 429.2,429.2, 464.2.464.2, 473.1.473.1, 492.9,492.9, 647.0647.0