15  3.3 Exercises

15.1 Exercises 3.3.1

For each of the following scenarios, state whether you think the null distribution of the statistic would be: almost perfectly Bell Curve shaped, approximately Bell Curve shaped, or not at all Bell Curve shaped.

  1. A sample of 18 people is taken, and their mean height is calculated.

  2. A coin is flipped 10 times, and the researcher records the percentage of “Heads” that results.

  3. A weather researcher records the wind speed and the number of inches of rain each day for a month (30 days). She then calculates the correlation between wind speed and inches of rain.

  4. A person wins the lottery, and purchases three vacation homes in California. He finds the average (mean) cost of his new houses.

  5. There are 500 students enrolled in the class “Introduction to Data Science”. The teacher records all of their Final Exam scores out of 100.

15.2 Exercises 3.3.2

For each of the following scenarios, use the Empirical Rule to approximate the p-value.

Your process should be:

  • Write down the null and alternate hypotheses.

  • Find the z-score.

  • Draw a sketch of the Standard Normal curve, draw a line for the z-score, and shade in the area under the curve that represents your p-value.

  • Use the Empirical Rule to estimate the area under the curve that you have shaded.

  1. A researcher is studying the average weight of grizzly bears in California versus in Alaska. She hypothesizes that Alaskan bears will be larger. She samples 30 bears from each location, and finds the following: \bar{x}_C = 425, \;\;\; s_C = 20.1 \bar{x}_A = 432, \;\;\; s_A = 15.4

  2. A weather researcher records the wind speed and the number of inches of rain each day for two months (60 days). She then calculates the correlation between wind speed and inches of rain, and finds a value of 0.31.

  3. A gambler is playing a very boring game at a casino: he rolls a die - where each number 1 to 6 is equally likely to show up - and if it is a 6 he wins a dollar. After playing 100 games, the gambler has only won $15. He suspects that the casino is cheating, and has dice that are designed to roll sixes less often.

15.3 Exercises 3.3.3

  1. Create a 95% confidence interval for Question (1) above.
  2. Create a 68% confidence interval for Question (2) above.
  3. Create a 99% confidence interval for Question (3) above.

15.4 Exercises 3.3.4

  1. Express the Null and Alternate Hypotheses in Question (1) above in terms of two generative models.
  2. Express the Null and Alternate Hypotheses in Question (3) above in terms of a generative model.