Consider an instrument that is used to measure temperature (°F) inside of an office building. Such an instrument hangs on a wall just outside of Dilbert's cubicle. Dilbert's office building is temperature-controlled, and set to a year-round mean temperature of 68°F, with a standard deviation of 6°F.
Dilbert found an article online that claims that uncomfortable temperatures in the workplace make unproductive workers. Specifically, it states that temperatures below 66°F or above 71°F have been shown to make employees uncomfortable -- sleepy and/or grumpy. Dilbert will be evaluated by his boss next month, and he now thinks that he will blame his lack of productivity on uncomfortable temperatures in the office. He decides that the next day he will meticulously check and record the temperature every 15 minutes for 8 hours. In this sample of size 32, he finds that the average temperature is 66.4°F with a standard deviation of 6.2.
1) Identify the given parameters, statistics and sample size. Use correct notation!
2) What is the sampling distribution of the average temperature for all such samples of size 32? Justify.
3) Compute the probability that the average temperature Dilbert will observe is considered uncomfortable (thus causing his lack of productivity.)
4) Dilbert doubts that the year-round mean temperature in the office is 68°F, as is claimed by the thermostat. He thinks that it must be lower, causing his excessive drowsiness. Test an appropriate hypothesis at the .05 level of significance. a. Write the null and alternative hypotheses.
b. State the null hypothesis in words.
c. What would be the implications of a Type I error in this case? What is the probability of making this type of error?
d. Compute the statistic and define the critical region.
e. What is your conclusion, and interpretation?
5) Provide a 95% confidence interval for the true mean temperature. (Assume that population parameters are unknown.) Interpret.
Determined to prove his point, Dilbert continues to collect data for a period of 30 workdays. He records each day as "comfortable" if the average temperature for the day is between 66 and 71 degrees, and "uncomfortable" otherwise. The probability that a day will be uncomfortable has already been computed in problem #3. In his sample of 30 days, only 2 days qualified as uncomfortable.
6) Identify the population and sample proportions of uncomfortable days, using correct notation.
7) Using sample data only, estimate the true proportion of uncomfortable days by computing a 95% confidence interval. Interpret. Does the CI include the parameter value?
8) During the evaluation, Dilbert tells his boss that his lack of productivity is due to uncomfortable temperatures in the office, which affect at least 15% of his work days. Using all the information above, do you think Dilbert makes a valid point? Explain your answer using any of the above computations, and discuss your concerns about the data provided.