Follow these steps when answering all confidence region questions:
  1. Identify the level of confidence (i.e., 100(1-α)%).
  2. Identify whether a lower confidence bound (greater than HA), upper confidence bound (less than HA), or confidence interval (not equals HA) should be constructed.
  3. Find Z* (Include a drawing to illustrate your calculation and your distrib() code).
  4. Calculate the confidence region values (i.e., x̄+Z*SE).
  5. Specifically interpret the confidence region with a complete sentence.

See the last section of the module reading and here for a demonstration of these steps.


Body Temperature

Machowiak et al. (1992)1 provided a critical examination of whether normal body temperature was 37oC. They recorded the orally-determined body temperatures of 65 male and 65 female subjects. Use their results in Table 1 and assume that α=0.05 and σ=0.45 to construct and interpret a confidence region for the population mean body temperature.

Table 1: Summary statistics for the body temperature of a sample of men and women.

     n   mean     sd    min     Q1 median     Q3    max 
130.00  36.81   0.41  35.70  36.60  36.80  37.10  38.20


Beetle Size

Researchers examined the size of two different species of beetles. They hypothesized that the thorax length of the Halticus oleracea species would be greater than 190 μm. Use their results for the Halticus oleracea species in Table 2 and assume that σ=15 and α=0.05 to construct and interpret a confidence region for the beetle’s mean thorax length.

Table 2: Summary statistics for the thorax length for two species of beetles.

             species  n   mean    sd min     Q1 median     Q3 max
1 Halticus.carduorum 20 179.55 10.09 158 175.75  180.5 181.75 198
2  Halticus.oleracea 18 194.17 14.03 170 189.75  192.0 200.75 221


Internet Usage

This is not a confidence region question, rather it is asking you to compute a sample size given σ, a margin-of-error tolerance, and a level of confidence (which, ultimately, is turned into a Z*). See here or the appropriate section in the reading for the formula and example calculations.

 

Suppose that you are starting a business and it is important for your business plan to have an estimate of the mean weekly Internet usage of households in your city. Assume that you know from previous surveys that the standard deviation of weekly Internet usage is 7.55 minutes.2

  1. How many households must you randomly select to be 95% sure that the sample mean is within 1 minute of the population mean.
  2. How many households must you randomly select to be 90% sure that the sample mean is within 1 minute of the population mean.
  3. How many households must you randomly select to be 95% sure that the sample mean is within 0.5 minutes of the population mean.

 

  1. This question was adapted from here.↩︎

  2. This question was adapted from here.↩︎