Question

Determine and interpret the linear correlation coefficient, and use linear regression to find a best fit line for a scatter plot of the data and make predictions. 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. Concepts 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.

MAG	DEPTH
0.70	7.2
0.74	2.2
0.64	13.9
0.39	15.5
0.70	3.0
2.20	2.4
1.98	14.4
0.64	5.7
1.22	6.1
0.20	9.1
1.64	17.2
1.32	8.7
2.95	9.3
0.90	12.3
1.76	7.4
1.01	7.0
1.26	17.1
0.00	8.8
0.65	6.0
1.46	19.1
1.62	12.7
1.83	4.7
0.99	8.6
1.56	6.0
0.40	14.6
1.28	4.9
0.83	19.1
1.34	9.9
0.54	16.1
1.25	4.6
0.92	4.9
1.00	16.1
0.79	14.0
0.79	4.2
1.44	5.9
1.00	5.4
2.24	15.6
2.50	7.7
1.79	15.4
1.25	16.4
1.49	4.9
0.84	8.1
1.42	7.5
1.00	14.1
1.25	11.1
1.42	16.0
1.35	5.5
0.93	7.3
0.40	3.1
1.39	6.0

Answer 1

1)Enter the data in EXCEL – select the data – select INSERT tab – select SCATTER in Chart – select the first type. The plot is shown in the picture.

2)The value of the correlation coefficient r = 0.0270197.
[Select DATA tab – Select DATA ANALYSIS – choose CORRELATION – select OK – select the entire data in the input range – tick the box if labels are present in the first row – grouped by COLUMNS – select OK. ]

Here, we want to test H₀ : = 0 ag. H₁ : 0, where = population correlation coefficient.
Test statistic : T = {r √(n-2)} / √(1 – r²) ~ t_n-2.
Critical value of the test : t _0.025,n-2 = 2.010635, where n=50, alpha=0.05.

3)Observed value of T = 0.187266.
Since observed T < critical value, we fail to reject H₀ at 5% level of significance and conclude that there is sufficient evidence to claim zero linear correlation between the magnitudes and the depths from the earthquakes.

4) [Select DATA tab – Select DATA ANALYSIS – choose REGRESSION – select OK – select the depth column in the input X range – select the magnitude column in the input Y range - tick the box if labels are present in the first row – select OK. ]

	Coefficients
Intercept	1.153154
DEPTH	0.003165

The regression equation is :
MAGNITUDE = 1.153154 + 0.003165 * DEPTH