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Scenario

According to the U.S. Geological Survey (USGS), the probability of a magnitude 6.7 or greater earthquake in the Greater Bay Area is 63%, about 2 out of 3, in the next 30 years. In April 2008, scientists and engineers released a new earthquake forecast for the State of California called the Uniform California Earthquake Rupture Forecast (UCERF).

As a junior analyst at the USGS, you are tasked to determine whether there is sufficient evidence to support the claim of a linear correlation between the magnitudes and depths from the earthquakes. Your deliverables will be a PowerPoint presentation you will create summarizing your findings and an excel document to show your work.

Concept being Studied

  • Correlation and regression
  • Creating scatterplots
  • Constructing and interpreting a Hypothesis Test for Correlation using r as the test statistic

You are given a spreadsheet that contains the following information:

  • Magnitude measured on the Richter scale
  • Depth in km

Using the spreadsheet, you will answer the problems below in a PowerPoint presentation.

What to Submit

The PowerPoint presentation should answer and explain the following questions based on the spreadsheet provided above.

  • Slide 1: Title slide
  • Slide 2: Introduce your scenario and data set including the variables provided.
  • Slide 3: Construct a scatterplot of the two variables provided in the spreadsheet. Include a description of what you see in the scatterplot.
  • Slide 4: Find the value of the linear correlation coefficient r and the critical value of r using α = 0.05. Include an explanation on how you found those values.
  • Slide 5: Determine whether there is sufficient evidence to support the claim of a linear correlation between the magnitudes and the depths from the earthquakes. Explain.
  • Slide 6: Find the regression equation. Let the predictor (x) variable be the magnitude. Identify the slope and the y-intercept within your regression equation.
  • Slide 7: Is the equation a good model? Explain. What would be the best predicted depth of an earthquake with a magnitude of 2.0? Include the correct units.
  • Slide 8: Conclude by recapping your ideas by summarizing the information presented in context of the scenario.

Along with your PowerPoint presentation, you should include your Excel document which shows all calculations.

Which of the following is not a correct value for a linear correlation coefficient for sample data r?

Select one:

a. 0.0012

b. 1/7

c. – 0.95

d. 1.0002

A correlation coefficient of -0.95 indicates what kind of relations between the two variables?

Select one:

a. Strong positive correlation

b. Weak negative correlation

c. Strong negative correlation

d. No correlation

Question 3

The relationship between coefficient of correlation and coefficient of determination is that:

Select one:

a. They are unrelated

b. The coefficient of determination is the coefficient of correlation squared

c. The coefficient of correlation is the coefficient of determination squared

d. They are equal

Question 4

When determining whether a correlation exists, it is a good idea to first explore the data by plotting a scatter plot.

Select one:

True

False

Question 5

a. The strength of correlation between the dependent and independent variables

b. The difference between two variables

c. Standard error of estimate

d. The percent of variations in the dependent variable explained by the independent variables

Question 6

Regression equations are often useful for predicting the value of one variable, given a value of the other variable.

Select one:

True

False

Question 7

Which one of the following values is not required to calculate the correlation coefficient r?

Select one:

a. The number of pairs of sample data n

b. The sum of all values of x Σx

c. The sum of all values of x2y2 Σx2y2

d. The sum of x multiplied by y Σxy

Question 8

The most commonly used formula to describe the linear regression is:

Select one:

a.

b.

c.

d.

Question 9

Which of the following is not a name for the straight line that best fits the scatter plot of paired sample data?

Select one:

a. Regression line

b. Line of best fit

c. Scatter line

d. Least-squares line

Question 10

A correlation exists between two variables only when the values of one variable are very strongly associated with the values of the other variable.

Select one:

True

False

Question 11

Which of the following is not a property of the linear correlation coefficient r?

Select one:

a. – 1 ≤ r ≤ 1

b. x and y are interchangeable

c. r is a measurement of the strength of a linear relationship

d. r is not sensitive to outliers

Question 12

If we determine that there is a correlation between poverty rate and crime rate in a city, then we can conclude that the increase in poverty causes people to commit more crime.

Select one:

True

False

Question 13

If the regression equation is not a good model, means there is no linear correlation, how can we use a sample to find the predicted value of y?

Select one:

a. Use the mean of the actual y values

b. Use the mode of the actual y values

c. Use the median of the actual y values

d. We cannot use sample data to make any predictions

Question 14

If the absolute value of correlation coefficient |r| is bigger than the critical value, which of the following conclusions is correct?

Select one:

a. There is no sufficient evidence to support the claim of a linear correlation.

b. There is sufficient evidence to support the claim of a linear correlation.

c. There may or may not be a linear correlation between the two variables.

d. There is sufficient evidence to support the claim of a non-linear correlation.

Question 15

When we interpret the determination coefficient r2, we are saying that

Select one:

a. For each unit increase in x, we will see an increase or decrease in the predicted variable y

b. The sample is significantly different from the population

c. There is a strong positive or negative relationship between the variables

d. Some portion of the dependent variable co-varies with some portion of the independent variable

Question 16

Predicted y = 20000 + 650x, where x = years of post-secondary educations and y = starting annual income. How is this regression equation interpreted?

Select one:

a. For every year increase in income, education increases by $650.

b. For every year increase in education, expected starting income increases by $650.

c. For every year increase in education, expected starting income decreases by $650.

d. If x were equal to zero, income would be predicted to be $650.

Question 17

When two variables are not related at all, how would you attach a quantitative measure to that situation?

Select one:

a. Correlation coefficient r<0

b. Correlation coefficient r≤0

c. Correlation coefficient r=0

d. No quantitative measure exists

Question 18

How will you construct a hypothesis test for correlation using r as the test statistic?

Select one:

a. H0: ρ = 0 (no correlation); Ha: ρ ≠ 0 (there is a correlation)

b. H0: r = 0 (no correlation); Ha: r ≠ 0 (there is a correlation)

c. H0: ρ≠ 0 (no correlation); Ha: ρ = 0 (there is a correlation)

d. H0: ρ≠ 0 (there is a correlation); Ha: ρ = 0 (no correlation)

Question 19

The value of determination coefficient r2 indicates the proportion of the variation in y that is explained by the linear relationship between x and y.

Select one:

True

False

Question 20

What is a correct conclusion when | r | ≤ critical value?

Select one:

a. Reject the null hypothesis and conclude that there is sufficient evidence to support the claim of a linear correlation.

b. Fail to reject the null hypothesis and conclude that there is no sufficient evidence to support the claim of a linear correlation.

c. Fail to reject the null hypothesis and conclude there is sufficient evidence to support the claim of a linear correlation.

d. Reject the null hypothesis and conclude that there is no sufficient evidence to support the claim of a linear correlation.

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