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The time it takes for a planet to complete its orbit around the sun is called the planet's sidereal year.

In 1618, Johannes Kepler discovered that the sidereal year of a planet is related to the distance the planet is from the sun.

The following data show the distances of the planets, and the dwarf planet Pluto, from the sun and their sidereal years

Planet

Distance from Sun,

x (millions of miles) Sidereal Year, y

Mercury 36, 0. 24

Venus 67, 0. 62

Earth 93, 1

Mars 142, 1. 88

Jupiter 483, 11. 9

Saturn 887, 29. 5

Uranus 1785, 84

Neptune 2797, 165

Pluto 3675, 248

(b) Determine the correlation between distance and sidereal year. Does this imply a linear relation between distance and sideral year?

User CMIVXX
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Answer: To determine the correlation between distance and sidereal year, we can calculate the correlation coefficient, often denoted by "r." The correlation coefficient quantifies the strength and direction of the linear relationship between two variables.

Before calculating the correlation coefficient, we need to standardize the data by converting distances from millions of miles to millions of kilometers. (1 mile ≈ 1.60934 km)

Updated Data:

Planet Distance from Sun (millions of km) Sidereal Year (years)

Mercury 36 * 1.60934 0.24

Venus 67 * 1.60934 0.62

Earth 93 * 1.60934 1

Mars 142 * 1.60934 1.88

Jupiter 483 * 1.60934 11.9

Saturn 887 * 1.60934 29.5

Uranus 1785 * 1.60934 84

Neptune 2797 * 1.60934 165

Pluto 3675 * 1.60934 248

Now, we can calculate the correlation coefficient (r) using the following formula:

r = [nΣ(xy) - ΣxΣy] / [√(nΣx^2 - (Σx)^2) √(nΣy^2 - (Σy)^2)]

where:

n = number of data points (in this case, 9 planets)

Σxy = sum of (distance from the sun * sidereal year) for all planets

Σx = sum of distance from the sun for all planets

Σy = sum of sidereal year for all planets

Σx^2 = sum of (distance from the sun)^2 for all planets

Σy^2 = sum of (sidereal year)^2 for all planets

Let's calculate the correlation coefficient:

n = 9

Σxy = (36 * 0.24) + (67 * 0.62) + (93 * 1) + (142 * 1.88) + (483 * 11.9) + (887 * 29.5) + (1785 * 84) + (2797 * 165) + (3675 * 248) ≈ 814811.75

Σx = 36 + 67 + 93 + 142 + 483 + 887 + 1785 + 2797 + 3675 ≈ 9885

Σy = 0.24 + 0.62 + 1 + 1.88 + 11.9 + 29.5 + 84 + 165 + 248 ≈ 542.14

Σx^2 = (36)^2 + (67)^2 + (93)^2 + (142)^2 + (483)^2 + (887)^2 + (1785)^2 + (2797)^2 + (3675)^2 ≈ 28343401

Σy^2 = (0.24)^2 + (0.62)^2 + (1)^2 + (1.88)^2 + (11.9)^2 + (29.5)^2 + (84)^2 + (165)^2 + (248)^2 ≈ 88326.58

Now, plug the values into the formula:

r = [9 * 814811.75 - (9885 * 542.14)] / [√(9 * 28343401 - (9885)^2) √(9 * 88326.58 - (542.14)^2)]

r ≈ 0.993

The correlation coefficient (r) is approximately 0.993. This indicates a strong positive correlation between the distance from the sun and the sidereal year. However, it's important to note that while the correlation is strong, it does not necessarily imply a linear relationship between distance and sidereal year. Other functional forms, such as exponential or power-law relationships, may also provide a good fit to the data.

User Kittsil
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