In January 2006, Your Family Moved To A Tropical Climate
In January Of 2006 Your Family Moved To A Tropical Climate For The Y
In January of 2006, your family moved to a tropical climate, and over the following year, you recorded the number of rainy days each month. The recorded data consists of the following numbers of rainy days per month: 14, 14, 10, 12, 11, 13, 11, 11, 14, 10, 13, 8. Your task involves calculating statistical measures—specifically, the mean, mode, median, and range—for this data set. Additionally, given a normal distribution with a mean of 16 and a standard deviation of 6, you are asked to find the Z-score corresponding with a value of 25.
Furthermore, you are required to solve a system of equations using the substitution method. The systems to solve are:
- 3x - 2y = 18
- 5x + 10y = -10
Paper For Above instruction
To analyze the rainy days recorded after moving to a tropical climate, it is essential to examine the data statistically. The dataset of monthly rainy days includes 14, 14, 10, 12, 11, 13, 11, 11, 14, 10, 13, and 8. Calculating the mean, median, mode, and range provides insight into the distribution and variability of rainy days, which can influence planning and resource allocation in the region.
Statistical Measures of the Data
The first step involves computing the mean, which is the average number of rainy days per month. Summing all values gives:
14 + 14 + 10 + 12 + 11 + 13 + 11 + 11 + 14 + 10 + 13 + 8 = 132
Since there are 12 months, the mean is:
Mean = 132 / 12 = 11
Next, the mode indicates the most frequently occurring value(s). In this data set, the number 11 appears three times, which is more than any other value, so the mode is:
Mode = 11
The median is the middle value when data points are ordered from smallest to largest. Sorting the data yields:
8, 10, 10, 11, 11, 11, 12, 13, 13, 14, 14, 14
With 12 data points, the median is the average of the 6th and 7th values:
Median = (11 + 12) / 2 = 11.5
The range measures the spread by subtracting the smallest value from the largest:
Range = 14 - 8 = 6
Calculating the Z-Score for a Given Value
The Z-score is a measure of how many standard deviations a specific value is from the mean in a normal distribution. The formula is:
Z = (X - μ) / σ
Where X is the value of interest, μ is the mean, and σ is the standard deviation. Given μ = 16 and σ = 6, for X = 25:
Z = (25 - 16) / 6 = 9 / 6 = 1.5
Thus, the Z-score corresponding to 25 is 1.5, indicating that 25 is 1.5 standard deviations above the mean.
Solving the System of Equations by Substitution
The two equations are:
- 3x - 2y = 18
- 5x + 10y = -10
To use substitution, solve the first equation for x:
3x = 18 + 2y → x = (18 + 2y) / 3
Substitute this expression for x into the second equation:
5 * [(18 + 2y) / 3] + 10y = -10
Multiply through to eliminate the denominator:
(5/3)(18 + 2y) + 10y = -10
Distribute:
(90/3) + (10/3)y + 10y = -10
Simplify:
30 + (10/3)y + 10y = -10
Expressing 10y as (30/3)y for common denominator:
30 + (10/3)y + (30/3)y = -10
Combine like terms:
30 + (40/3)y = -10
Subtract 30 from both sides:
(40/3)y = -40
Multiply both sides by 3 to clear denominator:
40y = -120
Divide both sides by 40:
y = -3
Now, substitute y = -3 back into the expression for x:
x = (18 + 2*(-3)) / 3 = (18 - 6) / 3 = 12 / 3 = 4
Hence, the solution to the system is:
x = 4, y = -3
Conclusion
The statistical analysis of the rainy days shows variability with a mean of 11 rainy days per month, a mode of 11, a median of 11.5, and a range of 6 days. The Z-score of 25 in a normal distribution with mean 16 and standard deviation 6 is 1.5, indicating that 25 is above average. The algebraic system yields the solution x = 4 and y = -3, demonstrating the effectiveness of the substitution method for solving such equations. Collectively, these calculations provide meaningful insights into the climate data and fundamental algebra techniques essential for analyzing environmental data and solving linear systems.
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