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 IIIIIIVRegion under Normal Curvex°FExpected % from Normal CurveObserved Number of Days in 20 Yearsμ – 3σ ≤ x < μ – 2σ 51 ≤ x < 592.35% 16 μ – 2σ ≤ x < μ – σ 59 ≤ x < 6713.5% 76 μ – σ ≤ x < μ 67 ≤ x < 7534% 209 μ ≤ x < μ + σ 75 ≤ x < 8334% 220 μ + σ ≤ x < μ + 2σ 83 ≤ x < 9113.5% 87 μ + 2σ ≤ x < μ + 3σ 91 ≤ x < 992.35% 12 1. Remember that μ = 75 and σ = 8. Examine the figure above. Write a brief explanation for columns I, II, and III in the context of this problem. 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.) What are the degrees of freedom? **Area Under a Normal Curve**The diagram represents the "Area Under a Normal Curve," which is also known as the "Empirical Rule" or "68-95-99.7 Rule." This rule describes how data is distributed in a normal (bell-shaped) distribution.The horizontal axis represents the variable of interest, while the vertical axis indicates probability density.Key components of the diagram include:1. **Mean (μ):** The center of the curve. This represents the average value in the dataset. 2. **Standard Deviation (σ):** Measures the amount of variation or dispersion in a set of values. From the mean, the normal distribution extends on both sides, approaching the horizontal axis asymptotically.**Regions Under the Curve:**- **Within ±1σ (from μ - 1σ to μ + 1σ):** - The central blue region covers this area. - Approximately 68% of the data falls within this range. - This region is composed of two parts: - From μ - 1σ to μ (left side of the mean): 34% - From μ to μ + 1σ (right side of the mean): 34%- **Within ±2σ (from μ - 2σ to μ + 2σ):** - The blue and light green regions combined cover this area. - Approximately 95% of the data falls within this range. - This region includes: - From μ - 2σ to μ - 1σ (left side): 13.5% - From μ + 1σ to μ + 2σ (right side): 13.5%- **Within ±3σ (from μ - 3σ to μ + 3σ):** - The blue, light green, and yellow regions combined cover this area. - Approximately 99.7% of the data falls within this range. - This region includes: - From μ - 3σ to μ - 2σ (left side): 2.35% - From μ + 2σ to μ + 3σ (right side): 2.35%These percentages visually demonstrate the Empirical Rule, helping to understand how data in a normal distribution behaves relative to the mean and standard deviations.

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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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%	76
μ – σ ≤ x < μ	67 ≤ x < 75	34%	209
μ ≤ x < μ + σ	75 ≤ x < 83	34%	220
μ + σ ≤ x < μ + 2σ	83 ≤ x < 91	13.5%	87
μ + 2σ ≤ x < μ + 3σ	91 ≤ x < 99	2.35%	12

1. Remember that μ = 75 and σ = 8. Examine the figure above. Write a brief explanation for columns I, II, and III in the context of this problem.

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.)

What are the degrees of freedom?

**Area Under a Normal Curve**

The diagram represents the "Area Under a Normal Curve," which is also known as the "Empirical Rule" or "68-95-99.7 Rule." This rule describes how data is distributed in a normal (bell-shaped) distribution.

The horizontal axis represents the variable of interest, while the vertical axis indicates probability density.

Key components of the diagram include:

1. **Mean (μ):** The center of the curve. This represents the average value in the dataset.

2. **Standard Deviation (σ):** Measures the amount of variation or dispersion in a set of values.

From the mean, the normal distribution extends on both sides, approaching the horizontal axis asymptotically.

**Regions Under the Curve:**

- **Within ±1σ (from μ - 1σ to μ + 1σ):**
- The central blue region covers this area.
- Approximately 68% of the data falls within this range.
- This region is composed of two parts:
- From μ - 1σ to μ (left side of the mean): 34%
- From μ to μ + 1σ (right side of the mean): 34%

- **Within ±2σ (from μ - 2σ to μ + 2σ):**
- The blue and light green regions combined cover this area.
- Approximately 95% of the data falls within this range.
- This region includes:
- From μ - 2σ to μ - 1σ (left side): 13.5%
- From μ + 1σ to μ + 2σ (right side): 13.5%

- **Within ±3σ (from μ - 3σ to μ + 3σ):**
- The blue, light green, and yellow regions combined cover this area.
- Approximately 99.7% of the data falls within this range.
- This region includes:
- From μ - 3σ to μ - 2σ (left side): 2.35%
- From μ + 2σ to μ + 3σ (right side): 2.35%

These percentages visually demonstrate the Empirical Rule, helping to understand how data in a normal distribution behaves relative to the mean and standard deviations.

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