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Question1. Given commuter and parking space data, find the correlation coefficient, critical values for rr, and check significance at 0.05.
2. For CPI and subway fare data, find the regression equation with CPI as xx and predict fare for CPI = 182.5.

Studdy Solution

STEP 1

Assumptions1. We have two sets of data the number of commuters and the number of parking spaces at different Metro-North railroad stations. . We are asked to find the correlation coefficient, critical values for r, determine if there significant linear correlation using a0.05 significance level, and state the conclusion.
3. We are also given paired values of the Consumer Price Index (CPI) and the cost of subway fare and asked to find the regression equation and the best predicted cost of subway fare when the CPI is182.5 (in the year2000).

STEP 2

First, we need to find the correlation coefficient for the number of commuters and the number of parking spaces. The formula for the correlation coefficient (r) isr=n(∑xy)−(∑x)(∑y)[n∑x2−(∑x)2][n∑y2−(∑y)2] r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} where- n is the number of pairs of data- x and y are the individual data points- ∑xy is the sum of the product of paired data- ∑x and ∑y are the sums of the individual data points- ∑x² and ∑y² are the sums of the squares of individual data points

STEP 3

Now, plug in the given values for x (commuters) and y (parking spaces) into the formula to calculate the correlation coefficient.

STEP 4

Calculate the correlation coefficient.

STEP 5

Next, we need to find the critical values for r. The critical value of r is determined by the sample size (n) and the significance level (0.05 in this case). It can be found in a statistical table or calculated using a statistical software.

STEP 6

Compare the calculated correlation coefficient with the critical value. If the absolute value of the correlation coefficient is greater than the critical value, there is a significant linear correlation.

STEP 7

State the conclusion based on the comparison in the previous step.

STEP 8

Next, we need to find the regression equation for the paired values of the Consumer Price Index (CPI) and the cost of subway fare. The formula for the regression equation isy=a+bx y = a + bx where- y is the dependent variable (subway fare) - x is the independent variable (CPI) - a is the y-intercept- b is the slope, which can be calculated asb=n(∑xy)−(∑x)(∑y)n∑x2−(∑x)2 b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}

STEP 9

Now, plug in the given values for x (CPI) and y (subway fare) into the formula to calculate the slope (b).

STEP 10

Calculate the slope (b).

STEP 11

Next, calculate the y-intercept (a) using the formulaa=yˉ−bxˉ a = \bar{y} - b\bar{x} where- yˉ\bar{y} is the mean of y- xˉ\bar{x} is the mean of x

STEP 12

Now, plug in the calculated slope and y-intercept into the regression equation.

STEP 13

Next, we need to find the best predicted cost of subway fare when the CPI is182.5. Plug this value into the regression equation and solve for y.

STEP 14

Calculate the best predicted cost of subway fare when the CPI is182..

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