Sugar beet growers are interested in realizing higher yields...

Sugar beet growers are interested in realizing higher yields and higher sucrose percentages from their crops. But do they go together? The data that follow are from the Montana sugar beet crop-values listed are by county, yield is tons per acre, and sucrose is percentage of sucrose. a. What, if any, relationship do you expect to find between the yield per acre and the sucrose percentage for sugar beets? b. Draw the scatter diagram for yield in tons per acre $(x)$ and sucrose percentage $(y)$ for the Montana data. Describe the relationship as seen on the scatter diagram. Is it what you anticipated? c. Find the linear correlation coefficient. d. At the 0.05 level of significance, is the linear correlation coefficient significantly different from zero? e. One of the ordered pairs appears to be outside of the pattern created by the other 10 ordered pairs. What effect do you think removal of this pair from the data values would have on (1) the appearance of the scatter diagram, (2) the linear correlation coefficient, and ( 3 ) the answer in part d? f. Remove Carbon County from the data and answer parts b-d. Compare the results with your answers to part e.

Sugar beet growers are interested in realizing higher yields and higher sucrose percentages from their crops. But do they go together? The data that follow are from the Montana sugar beet crop-values listed are by county, yield is tons per acre, and sucrose is percentage of sucrose.
a. What, if any, relationship do you expect to find between the yield per acre and the sucrose percentage for sugar beets?
b. Draw the scatter diagram for yield in tons per acre $(x)$ and sucrose percentage $(y)$ for the Montana data. Describe the relationship as seen on the scatter diagram. Is it what you anticipated?
c. Find the linear correlation coefficient.
d. At the 0.05 level of significance, is the linear correlation coefficient significantly different from zero?
e. One of the ordered pairs appears to be outside of the pattern created by the other 10 ordered pairs. What effect do you think removal of this pair from the data values would have on (1) the appearance of the scatter diagram, (2) the linear correlation coefficient, and ( 3 ) the answer in part d?
f. Remove Carbon County from the data and answer parts b-d. Compare the results with your answers to part e.

Key Concepts

Outliers and Their Effects

Outliers are data points that deviate substantially from the overall pattern in a dataset and can have a significant impact on statistical analyses. In correlation analysis, an outlier can distort the apparent relationship between variables, influencing both the scatter diagram and the computed correlation coefficient. Recognizing and addressing outliers is crucial for ensuring accurate interpretations of the data.

Scatter Diagram

A scatter diagram (or scatter plot) is a graphical tool used to display the relationship between two quantitative variables by plotting one variable along the x-axis and the other along the y-axis. This visual representation aids in observing patterns, trends, clustering, and potential anomalies in the data, and it is fundamental in exploratory data analysis.

Linear Correlation Coefficient

The linear correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two continuous variables. It is typically represented by the symbol r and ranges from -1 to 1. Values near -1 or 1 indicate a strong linear relationship (negative or positive, respectively), while values near 0 suggest little to no linear association.

Significance Testing of Correlation

Significance testing for the correlation coefficient involves evaluating whether the observed correlation is statistically different from zero, indicating a true relationship in the underlying population. This is typically performed using hypothesis testing, where a test statistic is calculated and compared to critical values at a predetermined significance level (such as 0.05), helping to determine if the correlation is due to chance.

Transcript

00:01 In question 41, we have a scenario where sugar beet growers are interested in realizing higher yields and higher sucrose percentages from their crops.

00:12 But we don't know whether these two variables are related.

00:16 So we have data from the montana sugar beet crop.

00:22 And the values are listed by county and yield is given in tons per acre and sucrose is given.

00:31 As a percentage of sucrose in the sugar.

00:36 So in the first part of the question, we told to tell what we expect to find about the yield per acre and the sucrose percentage for sugar bits.

00:49 I would expect that a greater yield per acre would correspond to a greater sucrose percentage.

00:58 But let's find out what the truth is from the scatter diagram.

01:04 So in the second part of the question we're going to be drawing a scatter diagram and on the y -axis we're going to have the sucrose percentage and on the x -axis we're going to have the tons per acre.

01:17 So to come up with a scatter diagram we need to select the two columns for yield and sucrose percentage and then we insert the scatter diagram.

01:30 So in this scatter diagram the the axis as such that on the horizontal axis, the x -axis, we have the yield per acre, and in the vertical axis, the y -axis, we have the sucrose percentage.

02:13 So when you look at the relationship between the sucrose percentage and the yield per acre, you realize that there seems to be a negative association, a negative relationship, which is contrary to what i had expected.

02:33 Adding a trend line shows that there is a negative linear relationship between the two variables.

02:42 Now we're going to find that linear correlation coefficient and to do that we're going to use a formula r.

02:51 So formula for r, so it's equal to corral correlation.

02:56 And select the first column and also the second column and include the region is 0 .68 -613 so this is to indicate that there's a negative negative there's a negative linear relationship between these two variables that when one variable increases the other variable decreases in part d of the question we're going to to be checking whether the linear correlation coefficient is significantly different from 0.

03:38 And this is going to be a one -tailed test because we found that our linear correlation coefficient is negative.

03:49 So we're going to test whether this linear correlation is significant or significantly different from 0.

03:57 So in that case we need to work out the following.

04:00 We need note n and we also work out the p value bounds.

04:16 So n is going to be equal to 11.

04:22 Therefore we shall have 9.

04:25 It is a free note 1.

04:31 It's one tailed test.

04:33 We're going to use the table 11 for the critical values.

04:37 Okay, so on this table we're going to be looking at the row for 9 degrees of freedom and we'll be locating 0 .68613.

04:51 So 686 belongs to this region between half of 0 .01 and 0 .02.

05:05 So the p value bounds would be 0 .005 and 0...