Let's go step by step to answer your questions regarding the dataset and the critical values provided in the image.

Part A: Scatter Plot

For this part, I will outline the steps you need to follow to create the scatter plot using the given $X$ and $Y$ values.

X-values:

$10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5$

Y-values:

$9.14, 8.15, 8.74, 8.77, 9.26, 8.11, 6.14, 3.09, 9.12, 7.26, 4.74$

1. Create a Scatter Plot:
   • Plot the $X$ values on the horizontal axis (independent variable) and the $Y$ values on the vertical axis (dependent variable).
   • For each $X_i$ and $Y_i$, mark a point at the corresponding coordinates.

You can use Excel, Google Sheets, or any graphing tool to plot this. The scatter plot will give a visual sense of whether the data suggests a linear relationship.

Part B: Find the Linear Correlation Coefficient

To compute the linear correlation coefficient ($r$), follow these steps:

1. Compute the means:

   • Mean of $X$: $\bar{X} = \frac{10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5}{11} = 9$
   • Mean of $Y$: $\bar{Y} = \frac{9.14 + 8.15 + 8.74 + 8.77 + 9.26 + 8.11 + 6.14 + 3.09 + 9.12 + 7.26 + 4.74}{11} = 7.5$

2. Calculate the covariance and variances: The linear correlation coefficient $r$ formula is: $r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}$ I'll calculate that for you below:

Using Python, let me compute the exact values:The linear correlation coefficient $r$ is approximately $0.816$.

Interpretation:

• Since $r = 0.816$, this indicates a strong positive linear relationship between the two variables $X$ and $Y$.

Hypothesis Test for Linear Correlation:

To check if this correlation is significant, we can compare it against a critical value at the $\alpha = 0.05$ significance level for a sample size $n = 11$.

From the Table of Critical Values provided in your image, for $n = 11$ and $\alpha = 0.05$, the critical value is approximately 0.602.

Since $r = 0.816$ is greater than the critical value $0.602$, we can reject the null hypothesis and conclude that there is sufficient evidence to support the claim of a linear correlation between the two variables.

Part C: Feature of the Data Missed Without the Scatter Plot

If we don't construct a scatter plot, we may miss patterns such as outliers or non-linear relationships. The scatter plot visually shows how the data points are distributed and whether any anomalies exist that may not be detected just by calculating the correlation coefficient.

Do you want more details or have any further questions? Here are some follow-up questions for deeper understanding:

1. How would you calculate the covariance of the two datasets $X$ and $Y$?
2. What is the effect of an outlier on the correlation coefficient?
3. How can you determine the slope of the line of best fit from the given data?
4. What does it mean if the correlation coefficient is close to 1 or -1?
5. What would happen to the correlation if one or two points in the dataset are altered significantly?

Tip: When computing correlations, always check for outliers or data anomalies using visual tools like scatter plots.