Question

1 bella recorded data and used her graphingcalculator to find the equation for the line of best fit. she then used the correlation coefficient to determine the strength of the linear fit. which correlation coefficient represents the strongest linear relationship?

1. 0.9
2. -0.8
3. 0.5
4. -0.3

2 analysis of data from a statistical study shows a linear relationship in the data with a correlation coefficient of -0.254. which statement best summarizes this result?

1. there is a strong positive correlation between the variables
2. there is a moderate positive correlation between the variables
3. there is a strong negative correlation between the variables
4. there is a moderate negative correlation between the variables.

3 the results of a linear regression are shown below.
$y = ax + b$
$a = -1.15785$
$b = 139.317772$
$r = -0.906557832$
$r^2 = 0.8038159461$
which phrase best describes the relationship between $x$ and $y$?

1. strong negative correlation
2. weak positive correlation
3. strong positive correlation
4. weak negative correlation

4 what is the correlation coefficient of the linear fit of the data shown below, to the nearest hundredth?

1. 1.00
2. -0.95
3. 0.93
4. -1.00

graph of a scatter plot with x-axis and y-axis, points plotted
5 the scatterplot below compares the number of bags of popcorn and the number of sodas sold at each performance of the circus over one week.
popcorn sales and soda sales
scatter plot with x-axis labeled popcorn sales and y-axis labeled soda sales, points plotted
which conclusion can be drawn from the scatterplot?

1. there is a negative correlation between popcorn sales and soda sales.
2. there is a positive correlation between popcorn sales and soda sales.
3. there is no correlation between popcorn sales and soda sales.
4. buying popcorn causes people to buy soda.

Explanation:

Step1: Identify the problem type

This problem involves correlation coefficients and scatterplots, which fall under Statistics (a subfield of Mathematics). We need to analyze each question based on the concepts of correlation (positive/negative, strong/moderate/weak) and scatterplot interpretation.

Step2: Question 1

Correlation coefficients range from -1 to 1. Closer to 1 or -1 means stronger relationship. 0.9 is close to 1 (strong positive), -0.3 (weak negative), 0.5 (moderate positive). So the strongest linear relationship is 0.9 (option 1).

Step3: Question 2

Correlation coefficient \( r = -0.254 \). Negative sign means negative correlation. The magnitude (0.254) is small, so it's a weak negative correlation? Wait, no: -0.254 is moderate? Wait, typically: \( |r| \) near 1: strong, near 0.5: moderate, near 0: weak. So -0.254 is weak negative? Wait, the options: 1) strong positive, 2) strong negative, 3) moderate positive, 4) moderate negative? Wait, no, the options given: 1) strong positive, 2) strong negative, 3) moderate positive, 4) moderate negative? Wait the original question: "Analysis of data... linear relationship with correlation coefficient of -0.254. Which statement best summarizes this result?" Wait, maybe I misread. Wait, the options: 1) strong positive, 2) strong negative, 3) moderate positive, 4) moderate negative? No, wait the user's image: "1) There is a strong positive correlation... 2) There is a strong negative correlation... 3) There is a moderate positive correlation... 4) There is a moderate negative correlation". Wait, -0.254: the sign is negative (so negative correlation), and the magnitude is 0.254, which is moderate? Wait, maybe the options are mislabeled? Wait, no, maybe I made a mistake. Wait, typically, \( |r| \geq 0.7 \) is strong, \( 0.3 \leq |r| < 0.7 \) is moderate, \( |r| < 0.3 \) is weak. So -0.254 is weak negative, but the options don't have weak. Wait, maybe the question's coefficient is -0.524? Wait, the user's image: "correlation coefficient of -0.254" or maybe -0.524? Wait, maybe a typo. Assuming it's -0.524 (closer to -0.5), then moderate negative. But if it's -0.254, then maybe the options are different. Wait, maybe the original question has -0.524. Alternatively, maybe the options are: 1) strong positive, 2) strong negative, 3) moderate positive, 4) moderate negative. So -0.254: negative, and moderate? No, 0.254 is weak. Wait, maybe the question is -0.524. Let's proceed.

Step4: Question 3

Given \( r = -0.906557832 \), \( r^2 = 0.8038159461 \). The correlation coefficient \( r \) is negative (so negative correlation) and \( |r| \approx 0.907 \), which is close to 1, so strong negative correlation (option 1).

Step5: Question 4

We need to find the correlation coefficient of the scatterplot. The scatterplot (from the image) shows a negative linear trend (as x increases, y decreases). The points are close to a line, so \( r \) is negative and close to -1? Wait, the options: 1) 1.00 (perfect positive), 2) 0.93 (positive), 3) -0.95 (negative, strong), 4) -1.00 (perfect negative). The scatterplot: looking at the graph, the points form a negative linear pattern, not perfect (so not -1.00), but close to -1, so -0.95 (option 3).

Step6: Question 5

Scatterplot: Popcorn Sales (x) and Soda Sales (y). The points show a positive trend (as popcorn sales increase, soda sales increase). So conclusion: 2) There is a positive correlation between popcorn sales and soda sales.

Answer:

1. 1) 0.9
2. 4) (Assuming -0.254 is moderate negative, or if it's -0.524, still moderate negative)
3. 1) strong negative correlation
4. 3) -0.95
5. 2) There is a positive correlation between popcorn sales and soda sales

(Note: For question 2, if the correlation is -0.254, it's weak negative, but since options don't have weak, maybe a typo. Assuming the intended coefficient is -0.524, then moderate negative (option 4).)