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Significance of Correlation & Confidence Interval

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a. The use of electrical stimulation (ES) to increase muscular strength is discussed in the Journal of Orthopedic and Sports Physical Therapy. Seventeen healthy volunteers participated in the experiment. Muscular strength, Y, was measured as a torquein foot-pounds, and ES, X, was measured in mA (microamps). The equation for the line of best fit is given as Y = 1.8X + 28.7, and the Pearson correlation coefficient was 0.61.a. Was the correlation coefficient significantly different from zero? Use α = 0.05.b. Predict the torque for a current equal to 50 mA.

b.
In the autumn of 2003, the National Safe Kids Campaign conducted a study of helmet use among children ages 5 to 14 who participate in wheeled sports. Data were collected from various sites across the United States that were designated as places where children often engage in wheeled sports. Activity, apparent gender, and estimated age were recorded for each rider, along with information on helmet use. It was found that, overall, 41% of children were wearing a helmet while participating in a wheeled sport.
a. Was this study an experiment or an observational study?
b. Identify the parameter of interest.
c. Identify the statistic and give its value.
d. Classify the four variables as numerical or attribute.

c.
A coin is flipped three times.
a. Draw a tree diagram that represents all possible outcomes.
b. Identify all branches that represent the event "exactly one head occurred."
c. Find the probability of "exactly one head occurred."

d.
A chemist is testing a newly proposed analytical method and decides to use the currently accepted method for comparison. She takes 12 specimens of unknown concentrate and determines the concentration of each specimen, using both the proposed method and the current method. Do these two samples represent dependent or independent samples? Explain.

e.
The weights of full boxes of a certain kind of cereal are normally distributed with a standard deviation of 0.27oz. A sample of 18 randomly selected boxes produced a mean weight of 9.87 oz.
a. Find the 95% confidence interval for the true mean weight of a box of this cereal.
b. Find the 99% confidence interval for the true mean weight of a box of this cereal.
c. What effect did the increase in the level of confidence have on the width of the confidence interval?

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Solution Summary

The solution provides step by step method for the preparation of tree diagram and calculation of test statistic for significance of correlation coefficient and confidence interval for population mean. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

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Correlation and Regression Questions

In this problem set you will get some practice performing a linear regression analysis. If you use Statdisk or Excel to perform any portion of these analyses, please include the results, label them, and refer to them accordingly in your interpretations.

Listed below are the overhead widths (in cm) of seals measured from photographs and the weights of the seals (in kg). The data are based on "Mass Estimation of Weddell Seals Using Techniques of Photogrammetry," by R. Garrott of Montana State University. The goal of the study is to explore the relationship between the overhead widths and the weights of the seals and to determine whether there is enough evidence to conclude that it is reasonable to use the overhead width to estimate the weight of a seal.

Overhead Width (cm) 7.2 7.4 9.8 9.4 8.8 8.4
Weight (kg) 116 154 245 202 200 191

1. Find the correlation coefficient and the critical value of r at the 5% significance level. Is there sufficient evidence to conclude that there is a linear relationship between the overhead width and the weight of the seals? Explain this using the value of the correlation coefficient and the critical value of r.

2. Find the explained variation. Explain the meaning of the explained variation in the context of this situation.

3. Find the unexplained variation. Explain the meaning of the unexplained variation in the context of this situation.

4. Find the total variation. Demonstrate and explain the relationship between the explained variation, the unexplained variation, and the total variation in the context of this situation.

5. Find the coefficient of determination. Demonstrate and explain the relationship between the explained variation, the total variation, and the coefficient of determination in the context of this situation. Explain the meaning of the coefficient of determination in the context of this situation.

6. Find the standard error of estimate.

7. Write the equation for the regression line. Explain the meaning of the slope of this line in the context of this situation. Find the predicted weight in kg of a seal given that the width from an overhead photograph is 9.0 cm.

8. Use the prediction interval spreadsheet to find both a 95% prediction interval estimate and a 95% confidence interval estimate of the weight in kg of a seal given that the width from an overhead photograph is 9.0 cm. Explain the meaning of these interval estimates in the context of this situation. Explain the difference between a 95% prediction interval estimate and a 95% confidence interval estimate for any given situation.

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