Question 1: Correlation

This question set uses the Gentoo.xlsx Download Gentoo.xlsxdata set. This file contains data collected from a representative sample of 123 Gentoo penguins.

Use correlation analysis to determine if there is a linear relationship between flipperlength (mm) and body mass (g).

Link to Gentoo.xlsx: https://1drv.ms/x/s!Aj_pQpiSrV7orFdPP-oQTkv4vbzQ?e=o1S9nV 

Question set 1:

In Minitab, make scatterplot (Graph > Scatterplot > Simple) to visualize the relationship between flipper length and body mass. (It does not matter which variable is X or Y.)

Use Insert > Image to show your graph.

Question set 1

Step 1a: Check assumptions

For correlation, there are two assumptions: (1) both variables are quantitative and (2) the relationship between them is linear. Are the assumptions met (per your scatterplot above)?

Explain your answer.

Question set 1

Step 1b: Write the hypotheses for Pearson's r

(Note: The null hypothesis for you is provided below, where the Greek letter rho, , is a measure of the population correlation between two variables.)

H₀: = 0 (i.e. there is no linear relationship)

H_a: _____

Question set 1

Step 2: Compute the test statistic (Pearson's r) and determine the p-value

• In Minitab, choose Stat > Basic Statistics > Correlation > select the two variables
• Click Results > select Pairwise correlation table > OK

Insert your Minitab results(specifically, the Pairwise Pearson Correlations table at the end of your Minitab results).

Per your Minitab results, what is the correlation test statistic (Pearson's r)? ________

Question set 1

Steps 3 and 4: Per your Minitab results, provide the p-value. Decide to reject or fail to reject the null hypothesis. Complete these steps by filling in the blanks:

Because the p-value is _________, which is _________ (less OR greater) than an alpha level of 0.05, we _________ (reject OR fail to reject) the null hypothesis.

Question set 1

Step 5: State a "real world" conclusion (given your answer to Step 4 above)

Specifically, is there evidence of a linear relationship between flipper length and body mass in the population of all Gentoo penguins?

Question set 1B:Compute the coefficient of determination (R²) for the relationship between flipper length and body mass.

Hint: The coefficient of determination = (Pearson's r)²

Question Set 2: Simple Linear Regression

This question set uses the same Gentoo penguins data set as above.

In Question Set 1, we used correlation analysis to determine if there is a linear relationship between two quantitative variables. Now, we will use regression analysis to BOTH determine if there is a linear relationship AND generate a predictive (regression) equation for two quantitative variables.

Use the five-step hypothesis testing procedure below to determine if bill depth (mm) can be used to predict body mass (g) in the population of all Gentoo penguins.

NOTE: For this question set, the variables of interest are body mass and bill depth (not flipper length).

Question set 2

Step 1a: Write the hypotheses

The null hypothesis is provided for you below.

H₀: ₁= 0 (i.e. the slope of the regression line is 0)

H_a: _____

Question set 2.1

Use Minitab to check the four assumptions, starting with the linearity assumption (for details, see https://online.stat.psu.edu/stat200/lesson/12/12.3/12.3.2Links to an external site.):

For the linearity assumption, use Graph > Scatterplot > Simple > Y variable: body_mass_g and X variable: bill_depth_mm. Use Insert > Image to show your scatterplot.

Per the scatterplot, we want to see a linear relationship between the two variables. Is this assumption met? Explain your answer.

Question set 2.2

For the independence of errors assumption, use Stat > Regression > Regression > Fit Regression Model > select the Response:body_mass_g and Continuous predictor:bill_depth_mm > click Graphs > Select 'Residuals versus fits'. Insert the plot.

Per the plot, we want to see a random scatter of points. We do not want the residuals and fitted (i.e. predicted) values to be correlated? Is this assumption met? Explain your answer.

Question set 2.3

For the equal variances assumption, refer to your 'residuals versus fits' plot above.

Per the plot, we want to see that the variability in the residuals is consistent across the x-axis. Is this assumption met? Explain your answer.

Question set 2.4

For the normality of errors assumption, use Stat > Regression > Regression > Fit Regression Model > Graphs > Select 'Histogram of residuals'.Insert the plot.

Per the plot, we want the residuals to be approximately bell-shaped (i.e. normally distributed). Is this assumption met? Explain your answer.

Question set 2

Step 2: Compute the test statistic and determine the p-value

In Minitab, choose Stat > Regression > Regression > Fit Regression Model > select the Response: body_mass_g and Continuous predictor: bill_depth_mm.

Insert your Minitab results(specifically, the'Regression Equation,' 'Coefficients' and 'Model Summary').

Per your Minitab results, what is the T test statistic for bill_depth_mm? ________

Question set 2

Steps 3 and 4: Per your Minitab results, provide the p-value. Decide to reject or fail to reject the null hypothesis. Complete these steps by filling in the blanks:

Because the p-value is _________, which is _________ (less OR greater) than an alpha level of 0.05, we _________ (reject OR fail to reject) the null hypothesis.

Question set 2

Step 5: State a "real world" conclusion

Specifically, is there evidence that bill depth can be used to predict body mass,in the population of all Gentoo penguins?

Question set 2B:

The first penguin in this sample had a bill depth of 13.2 mm. Using the regression equation you provided above, compute the predicted body mass for this penguin. Show your work.

Question set 2C:

The first penguin in this sample had an actual body mass of 4500 grams. Using this value and the predicted value you computed in the previous question, compute the residual for the first penguin in this sample.

Hint: residual = actual body mass in the data set - predicted body mass