Let x be a random variable that represents the average daily temperature (in degrees Fahrenheit) in January for a town in Hawaii. The x variable has a mean ? of approximately 68°F and standard deviation ? of approximately 4°F. A 20-year study (620 January days) gave the entries in the rightmost column of the following table. I II III IV Region under Normal Curve x°F Expected % from Normal Curve Observed Number of Days in 20 Years ? − 3? ≤ x < ? − 2? 56 ≤ x < 60 2.35% 16 ? − 2? ≤ x < ? − ? 60 ≤ x < 64 13.5% 82 ? − ? ≤ x < ? 64 ≤ x < 68 34% 209 ? ≤ x < ? + ? 68 ≤ x < 72 34% 217 ? + ? ≤ x < ? + 2? 72 ≤ x < 76 13.5% 87 ? + 2? ≤ x < ? + 3? 76 ≤ x < 80 2.35% 9 (i) Remember that ? = 68 and ? = 4. Examine the Normal Distribution Curve. Write a brief explanation for columns I, II, and III in the context of this problem
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Let x be a random variable that represents the average daily temperature (in degrees Fahrenheit) in January for a town in Hawaii. The x variable has a
| I | II | III | IV |
| Region under Normal Curve | x°F | Expected % from Normal Curve | Observed Number of Days in 20 Years |
|
? − 3? ≤ x < ? − 2?
|
56 ≤ x < 60
|
2.35% | 16 |
|
? − 2? ≤ x < ? − ?
|
60 ≤ x < 64
|
13.5% | 82 |
|
? − ? ≤ x < ?
|
64 ≤ x < 68
|
34% | 209 |
|
? ≤ x < ? + ?
|
68 ≤ x < 72
|
34% | 217 |
|
? + ? ≤ x < ? + 2?
|
72 ≤ x < 76
|
13.5% | 87 |
|
? + 2? ≤ x < ? + 3?
|
76 ≤ x < 80
|
2.35% | 9 |
(i) Remember that ? = 68 and ? = 4. Examine the
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