Let x be a random variable that represents the average daily temperature (in degrees Fahrenheit) in July in a town in Colorado. The x distribution has a mean μ of approximately 75°F and standard deviation σ of approximately 8°F. A 20-year study (620 July 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 μ – 3σ ≤ x < μ – 2σ 51 ≤ x < 59 2.35% 16 μ – 2σ ≤ x < μ – σ 59 ≤ x < 67 13.5% 83 μ – σ ≤ x < μ 67 ≤ x < 75 34% 205 μ ≤ x < μ + σ 75 ≤ x < 83 34% 224 μ + σ ≤ x < μ + 2σ 83 ≤ x < 91 13.5% 79 μ + 2σ ≤ x < μ + 3σ 91 ≤ x < 99 2.35% 13 (ii) Use a 1% level of significance to test the claim that the average daily July temperature follows a normal distribution with μ = 75 and σ = 8. (b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)
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 July in a town in Colorado. The x distribution has a mean μ of approximately 75°F and standard deviation σ of approximately 8°F. A 20-year study (620 July 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 |
| μ – 3σ ≤ x < μ – 2σ | 51 ≤ x < 59 | 2.35% | 16 |
| μ – 2σ ≤ x < μ – σ | 59 ≤ x < 67 | 13.5% | 83 |
| μ – σ ≤ x < μ | 67 ≤ x < 75 | 34% | 205 |
| μ ≤ x < μ + σ | 75 ≤ x < 83 | 34% | 224 |
| μ + σ ≤ x < μ + 2σ | 83 ≤ x < 91 | 13.5% | 79 |
| μ + 2σ ≤ x < μ + 3σ | 91 ≤ x < 99 | 2.35% | 13 |
(ii) Use a 1% level of significance to test the claim that the average daily July temperature follows a
(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)
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