A student wonders if tall women tend to date taller men than do short women. She measures herself, her dormitory roommate, and the women in the adjoining rooms; then she measures the next man each woman dates. Here are the data (heights in inches):

Men (x) 67 62.1 66.9 63.3 72.7 62.1
Women (y) 74.1 67.2 69.7 69.1 71.6 63.1



Find the correlation between the heights of men and women (round your answer to at least two decimal places.)



How does the correlation change if all the men were 4 inches shorter than the heights given in the table?

Decrease by 4/

Increase by 1/4

Stay the same

Decrease by 1/4

Impossible to tell without plotting the new data.

Increase by 4/.



If heights were measured in centimeters rather than inches, what would be the correlation (± 0.01)? (There are 2.54 centimeters in an inch.)

help me please GUys

To find the correlation between the heights of men and women, we can use the Pearson correlation coefficient formula. The formula is:

r = (Σ [(x - x̄)(y - ȳ)]) / √[(Σ (x - x̄)²)(Σ (y - ȳ)²)]

Where:
- r is the correlation coefficient
- x and y are the individual values of the heights of men and women
- x̄ and ȳ are the means of the heights of men and women
- Σ denotes the sum of

Let's calculate the correlation coefficient using the given heights:

Men (x): 67, 62.1, 66.9, 63.3, 72.7, 62.1
Women (y): 74.1, 67.2, 69.7, 69.1, 71.6, 63.1

1. Calculate the means of x and y:
x̄ = (67 + 62.1 + 66.9 + 63.3 + 72.7 + 62.1) / 6 = 66.83
ȳ = (74.1 + 67.2 + 69.7 + 69.1 + 71.6 + 63.1) / 6 = 69.83

2. Calculate the deviations from the means:
x - x̄: -0.83, -4.73, 0.07, -3.53, 5.87, -4.73
y - ȳ: 4.27, -2.63, -0.13, -0.73, 1.77, -6.73

3. Square the deviations:
(x - x̄)²: 0.69, 22.35, 0.005, 12.46, 34.36, 22.35
(y - ȳ)²: 18.21, 6.92, 0.02, 0.53, 3.14, 45.36

4. Calculate the sums of the squared deviations:
Σ (x - x̄)² = 0.69 + 22.35 + 0.005 + 12.46 + 34.36 + 22.35 = 92.17
Σ (y - ȳ)² = 18.21 + 6.92 + 0.02 + 0.53 + 3.14 + 45.36 = 74.18

5. Calculate the sum of the products of deviations:
Σ [(x - x̄)(y - ȳ)] = (-0.83 * 4.27) + (-4.73 * -2.63) + (0.07 * -0.13) + (-3.53 * -0.73) + (5.87 * 1.77) + (-4.73 * -6.73) = 69.79

6. Calculate the correlation coefficient:
r = (Σ [(x - x̄)(y - ȳ)]) / √[(Σ (x - x̄)²)(Σ (y - ȳ)²)]
r = 69.79 / √(92.17 * 74.18)
r ≈ 0.99 (rounded to two decimal places)

Therefore, the correlation between the heights of men and women is approximately 0.99.

Now, let's consider how the correlation changes if all the men were 4 inches shorter than the given heights. We will subtract 4 from each value of x.

New Men (x): 63, 58.1, 62.9, 59.3, 68.7, 58.1

By changing the heights of men, the correlation coefficient will also change. However, since we do not have the new heights of women, it is impossible to determine precisely how the correlation changes without plotting the new data. Therefore, the answer is "Impossible to tell without plotting the new data."

If heights were measured in centimeters rather than inches, the correlation coefficient would remain the same. The correlation coefficient does not depend on the units of measurement; it is a measure of the linear relationship between variables, regardless of the scale.

To find the correlation between the heights of men and women, we can use the formula for the Pearson correlation coefficient. The formula is:

r = (n∑xy - (∑x)(∑y)) / sqrt((n∑x^2 - (∑x)^2)(n∑y^2 - (∑y)^2))

where:
n is the number of data points
∑ represents the sum of the values

Using the given data, we can calculate the correlation between the heights of men and women.

x: 67, 62.1, 66.9, 63.3, 72.7, 62.1
y: 74.1, 67.2, 69.7, 69.1, 71.6, 63.1

n = 6
∑x = 67 + 62.1 + 66.9 + 63.3 + 72.7 + 62.1 = 394.1
∑y = 74.1 + 67.2 + 69.7 + 69.1 + 71.6 + 63.1 = 414.8
∑xy = (67 * 74.1) + (62.1 * 67.2) + (66.9 * 69.7) + (63.3 * 69.1) + (72.7 * 71.6) + (62.1 * 63.1) = 26800.66
∑x^2 = (67^2) + (62.1^2) + (66.9^2) + (63.3^2) + (72.7^2) + (62.1^2) = 25332.29
∑y^2 = (74.1^2) + (67.2^2) + (69.7^2) + (69.1^2) + (71.6^2) + (63.1^2) = 28215.74

Now we can substitute these values into the formula for the correlation coefficient:

r = (6 * 26800.66 - (394.1)(414.8)) / sqrt((6 * 25332.29 - (394.1)^2)(6 * 28215.74 - (414.8)^2))

After calculating, the correlation between the heights of men and women is approximately 0.89.

Next, let's consider how the correlation changes if all the men were 4 inches shorter than the original heights. In this case, we would subtract 4 from each height measurement.

New x: 63, 58.1, 62.9, 59.3, 68.7, 58.1

Using the same formula as before, we can calculate the correlation coefficient for the new data set. However, it is not possible to determine the exact change in correlation without calculating it first.

Now, let's determine the correlation if heights were measured in centimeters instead of inches. We know that there are 2.54 centimeters in an inch. So, we can convert the original data to centimeters by multiplying each height measurement by 2.54.

Converted x: 170.18, 157.734, 169.926, 160.782, 184.902, 157.734
Converted y: 188.214, 170.688, 177.038, 175.514, 181.744, 160.274

Using the same formula as before, we can substitute the converted values into the formula to find the correlation coefficient. The calculation will give the correlation between the heights of men and women in centimeters.