Math Problem Statement

Listed below are the overhead widths​ (in cm) of seals measured from photographs and the weights​ (in kg) of the seals. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the critical values of r using alphaαequals=0.010.01. Is there sufficient evidence to conclude that there is a linear correlation between overhead widths of seals from photographs and the weights of the​ seals? Overhead Width Weight 7.2 114 7.6 192 9.8 249 9.3 197 8.8 203 8.4 195

Critical Values for the Correlation Coefficient n alpha = .05 alpha = .01 4 0.95 0.99 5 0.878 0.959 6 0.811 0.917 7 0.754 0.875 8 0.707 0.834 9 0.666 0.798 10 0.632 0.765 11 0.602 0.735 12 0.576 0.708 13 0.553 0.684 14 0.532 0.661 15 0.514 0.641 16 0.497 0.623 17 0.482 0.606 18 0.468 0.59 19 0.456 0.575 20 0.444 0.561 25 0.396 0.505 30 0.361 0.463 35 0.335 0.43 40 0.312 0.402 45 0.294 0.378 50 0.279 0.361 60 0.254 0.33 70 0.236 0.305 80 0.22 0.286 90 0.207 0.269 100 0.196 0.256 Note: To test H subscript 0: rho equals 0 against H subscript 1: rho not equal to 0, reject H subscript 0 if the absolute value of r is greater than the critical value in the table.

The linear correlation coefficient is __?

Solution

To analyze the relationship between the overhead widths and weights of seals, we need to:

  1. Construct a scatterplot.
  2. Calculate the linear correlation coefficient rr.
  3. Determine the critical value for rr given α=0.01\alpha = 0.01.
  4. Conclude if there is sufficient evidence of a linear correlation.

Step 1: Constructing a Scatterplot

Let's plot the given data points:

  • Overhead Width: {7.2,7.6,9.8,9.3,8.8,8.4}\{7.2, 7.6, 9.8, 9.3, 8.8, 8.4\}
  • Weight: {114,192,249,197,203,195}\{114, 192, 249, 197, 203, 195\}

Step 2: Calculate the Linear Correlation Coefficient rr

The formula for the Pearson correlation coefficient rr is:

[ 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:

  • nn is the number of data points.
  • xx and yy are the individual sample points.

Let's first compute the necessary sums:

x & y & xy & x^2 & y^2 \\ \hline 7.2 & 114 & 820.8 & 51.84 & 12996 \\ 7.6 & 192 & 1459.2 & 57.76 & 36864 \\ 9.8 & 249 & 2440.2 & 96.04 & 62001 \\ 9.3 & 197 & 1832.1 & 86.49 & 38809 \\ 8.8 & 203 & 1786.4 & 77.44 & 41209 \\ 8.4 & 195 & 1638.0 & 70.56 & 38025 \\ \hline \sum & & 9976.7 & 440.13 & 229904 \\ \end{array}$$ Now, calculate the sums: $$\begin{align*} \sum x &= 7.2 + 7.6 + 9.8 + 9.3 + 8.8 + 8.4 = 51.1 \\ \sum y &= 114 + 192 + 249 + 197 + 203 + 195 = 1150 \\ \sum xy &= 9976.7 \\ \sum x^2 &= 440.13 \\ \sum y^2 &= 229904 \\ n &= 6 \\ \end{align*}$$ Plug these values into the formula: \[ r = \frac{6(9976.7) - (51.1)(1150)}{\sqrt{[6(440.13) - (51.1)^2][6(229904) - (1150)^2]}} \] \[ r = \frac{59860.2 - 58765}{\sqrt{[2640.78 - 2611.21][1379424 - 1322500]}} \] $$r = \frac{1095.2}{\sqrt{29.57 \times 56924}}$$ $$r = \frac{1095.2}{\sqrt{1682340.68}}$$ $$r = \frac{1095.2}{1296.25}$$ $$r \approx 0.8449$$ ### Step 3: Determine the Critical Value for $$r$$ Given $$\alpha = 0.01$$ and $$n = 6$$, the critical value from the table is 0.917. ### Step 4: Conclusion The calculated correlation coefficient $$r \approx 0.8449$$ is less than the critical value of 0.917. Therefore, we do not reject the null hypothesis $$H_0: \rho = 0$$. ### Conclusion There is not sufficient evidence at the $$\alpha = 0.01$$ level to conclude that there is a linear correlation between the overhead widths of seals from photographs and their weights. Would you like more details or have any questions? Here are 8 related questions for further exploration: 1. How would the analysis change if the sample size increased? 2. What are the assumptions of Pearson's correlation coefficient? 3. How do outliers affect the correlation coefficient? 4. Can we use a different significance level to test the correlation? 5. How is the scatterplot useful in visualizing the data relationship? 6. What is the difference between correlation and causation? 7. Are there other methods to measure the relationship between two variables? 8. How can we interpret a correlation coefficient close to 1 or -1? **Tip:** Always plot your data first. Visualization can help detect patterns, trends, and potential outliers that could influence your statistical analysis.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation

Formulas

Pearson correlation coefficient

Theorems

Critical values for correlation coefficient

Suitable Grade Level

Higher Education