11 **Educational Content: Correlation Analysis Between Internet Users and Scientific Award Winners**This educational content is centered around analyzing the relationship between the number of Internet users per 100 people and the number of scientific award winners per 10 million people across different countries.**Data Given:**- Internet Users (per 100 people): 80.9, 78.1, 57.1, 67.1, 78.3, 38.3- Award Winners (per 10 million people): 5.6, 8.6, 3.2, 1.7, 11.2, 0.1**Objective:**1. Construct a scatterplot.2. Calculate the linear correlation coefficient \( r \).3. Determine the P-value of \( r \).4. Assess if there is enough evidence to support a linear correlation between the variables using a significance level of \( \alpha = 0.01 \).**Scatterplot Options:**Four scatterplot options (A, B, C, D) are provided, each showing a plot for Internet Users (x-axis: 30 to 90) versus Award Winners (y-axis: 0 to 12). Individuals must choose the correct scatterplot capturing the data trend.**Analysis Steps:**1. **Calculate the Linear Correlation Coefficient (\( r \)):** A measure of the strength and direction of a linear relationship between two variables. It should be calculated and rounded to three decimal places. 2. **Hypothesis Testing:** - Null Hypothesis (\( H_0 \)): No linear correlation (\( \rho = 0 \)) - Alternative Hypothesis (\( H_1 \)): Linear correlation exists (\( \rho \neq 0 \))3. **Test Statistic (\( t \)):** Calculate this statistic and round to two decimal places to test if the correlation is significantly different from zero.4. **P-value:** Determine the P-value related to the test statistic, rounded to three decimal places.5. **Conclusion Using Significance Level:** Evaluate the P-value against the significance level (\( \alpha = 0.01 \)) to decide whether there is sufficient evidence to claim a significant linear correlation.Use this guide to apply statistical concepts in exploring the relationship between digital connectivity and scientific achievement across nations.

Scatter plot is the diagrammatic representation of the bivariate data where one variable is plotted on the x-axis and another is plotted on the y-axis and the diagram of dots obtained is called scatter plot. Let X and Y be the two random variable, then the correlation coefficient between X and Y is defined as the numerical measure of linear relationship between them. It is denoted by r. Mathematically, r=∑ixi-x¯yi-y¯∑ixi-x¯2∑iyi-y¯2 Here, x¯=1n∑ixi and y¯=1n∑iyi. To test for the significance for correlation coefficient following procedure will be used: Test statistics: t=rn-21-r2 The test statistics follows the student's t-distribution with (n-2) degree of freedom. Compute P-value corresponding to calculated t-value and compare it with given level of significance., we will accept or reject the null hypothesis accordingly.For the given question, let X denotes 'Internet users' and Y denotes 'Award Winners'. For Scatterplot: We will plot the values of Internet users on the x-axis and the values of Award Winners will be plotted on the y-axis and the diagram of dots obtained is the required scatter plot, as shown below: This scatterplot resembles with option (D.). Therefore, the correct option is (D.).For correlation coefficient: The calculation table for correlation coefficient is as follows: xi yi xi-x¯2 yi-y¯2 xi-x¯yi-y¯ 80.9 5.6 203.5378 0.2844 7.6089 78.1 8.6 131.4844 12.4844 40.5156 57.1 3.2 90.8844 3.4844 17.7956 67.1 1.7 0.2178 11.3344 -1.5711 78.3 11.2 136.1111 37.6178 71.5556 38.3 0.1 802.7778 24.6678 140.7222 Total 399.8 30.4 1365.0133 89.8733 276.6267 Here, sample size (n) is 6.Using, table values, we get The mean for X and Y are calculated as follows: x¯=1n∑ixi=399.86=66.6333y¯=1n∑iyi=30.46=5.0667 The correlation coefficient is calculated as follows: r=∑ixi-x¯yi-y¯∑ixi-x¯2∑iyi-y¯2=276.62671365.013389.8733=276.626736.9461×9.4802=276.6267350.2546=0.7898 Therefore, the blank should be filled with 0.790 (rounded to three decimal place).For testing of hypothesis: Let ρ denotes the population correlation coefficient. For testing the claim that there exists a linear correlation between X and Y, the hypothesis will be as follows: We know that the null hypothesis of no difference, we will assume that r is not significant of correlation of the population, i.e., ρ=0. The alternative hypothesis is complement of null hypothesis so, we will assume that r is significant of correlation of the population, i.e., ρ≠0. Therefore, for null hypothesis, the blank should be filled with '=' and '0'. For alternative hypothesis, the blank should be filled with '≠' and '0'. The test statistics is calculated as: t=rn-21-r2=0.7898×6-21-0.78982=0.7898×41-0.6238=0.7898×20.3762=1.57960.6134=2.5752 Therefore, the blank should be filled with 2.58 (rounded to two decimal place).The P-value corresponding to calculated t-value (2.5752) at (6-2=4) degree of freedom and 0.01 level of significance is 0.06164. Therefore, the blank should be filled with 0.062 (rounded to three decimal place). Since P-value is greater than 0.01 level of significance so, we cannot reject the null hypothesis. Therefore, we can conclude that ρ=0 and the given claim is true. Therefore, the blank should be filled with 'greater than' and 'is' .

Math

Statistics

sted below are numbers of Internet users per 100 people and numbers of scientific award winners per 10 million people for different countries. Construct a scatterplot, find the value of the linear co nether there is sulficient evidence to support a claim of linear correlation between the two variables. Use a significance level of a =0.01. Internet Users Award Winners 80.9 78.1 38.3 O 0.1 57.1 67.1 78.3 112 5.6 8.6 3.2 1.7 onstruct a scatterplot. Choose the correct graph below. DA. O B. OC. 12 12 12 90 Intemet Users Intermet Users Intermet Uhers The linear correlation coefficient is r= Acerd Winnes UM PR UM ARY