Math Problem Statement

Yearly rainfall amounts (in millimeters) of a city are in the table.

Yearly Rainfall Amounts

113.9 61.9 97.9 72.6 186.2 231.5 210.5 207.6 247.4 177.4 221.8 95.6 89.0 167.9 74.5 88.8 43.3 47.6 144.4 115.7 90.0 140.8 100.5 28.7 43.7 180.7 43.0 145.8 100.4 121.4 71.8 6.5 150.5 219.4 194.4 105.6 133.5 174.8 184.9 150.3 100.5 43.3 200.1 226.0 46.7 160.8 136.1

Using technology, find the mean and standard deviation. Round to the nearest hundredth, if necessary. The mean of the data is mm.

The standard deviation of the data is mm.

Use Chebyshev’s theorem to find an interval centered about the mean yearly rainfalls in which you would expect at least 75% of the yearly rainfalls to fall.

At least 75% of the yearly rainfalls will fall between mm and mm.

Use Chebyshev’s theorem to find an interval centered about the mean yearly rainfalls in which you would expect at least 88.9% of the yearly rainfalls to fall.

At least 88.9% of the yearly rainfalls will fall between mm and mm.

Solution

The mean yearly rainfall is approximately 127.57 mm, and the standard deviation is about 62.31 mm.

Chebyshev's Theorem Intervals:

  1. At least 75% of the yearly rainfalls will fall between 2.95 mm and 252.19 mm.
  2. At least 88.9% of the yearly rainfalls will fall between -59.36 mm and 314.50 mm.

Would you like more details or clarification on these results?

Here are some related questions for further exploration:

  1. How does Chebyshev’s theorem work with different confidence levels?
  2. What would the intervals be if we wanted to capture 90% of the data?
  3. How does Chebyshev's theorem compare to the empirical rule?
  4. Can we identify any outliers in this data using the calculated intervals?
  5. How might seasonal variations influence yearly rainfall data?

Tip: Chebyshev's theorem applies to any distribution, regardless of shape, making it useful when the data is not normally distributed.

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Math Problem Analysis

Mathematical Concepts

Statistics
Mean
Standard Deviation
Chebyshev's Theorem

Formulas

Mean formula: (sum of values) / (number of values)
Standard deviation formula: sqrt((Σ(xi - μ)^2) / N)
Chebyshev's Theorem: (1 - (1/k^2)) * 100%

Theorems

Chebyshev's Theorem

Suitable Grade Level

Grades 10-12

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