# Continuous Probabilities: Normal Distributions

Task Background: In this week's discussion, you learned how to construct probability distributions and graph them. This week, you will review continuous probabilities, more specifically normal distributions.

You are hired as a statistical analyst for Silver's Gym, and your boss wants to examine the relationship between body fat and weight in men who attend the gym. After compiling the data for weight and body fat of 252 men who attend Silver's Gym, you find it relevant to examine the statistical measures and to perform hypothesis tests and regression analysis to help make general conclusions for body fat and weight in men.

Part I: Statistical Measures

Statistics is a very powerful topic that is used on a daily basis in many situations. For example, you may be interested in the age of the men who attend Silver's Gym. You could not assume that all men are the same age. Thus, it would be an inaccurate measure to state that "the average age of men who attend Silver's Gym is the same age as me."

Averages are only one type of statistical measurements that may be of interest. For example, your company likes to gauge sales during a certain time of year and to keep costs low to a point that the business is making money. These various statistical measurements are important in the world of statistics because they help you make general conclusions about a given population or sample.

To assist in your analysis for Silver's Gym, answer the following questions about the Body Fat Versus Weight data set:

Click here to download the Body Fat Weight data set.

Calculate the mean, median, range, and standard deviation for the Body Fat Versus Weight data set. Report your findings, and interpret the meanings of each measurement. Notice you are to calculate the mean, median, range, and standard deviation for the body fat and for the weight.

The measures of central tendency are important in real-world situations.

What is the importance of finding the mean/median? Why might you find this information useful?

In some data sets, the mean is more important than the median. For example, you want to know your mean overall grade average because the median grade average would be meaningless. However, you might be interested in a median salary to see the middle value of where salaries fall. Explain which measure, the mean or the median, is more applicable for this data set and this problem.

What is the importance of finding the range/standard deviation? Why might you find this information useful?

Part II: Hypothesis Testing

Organizations sometimes want to go beyond describing the data and actually perform some type of inference on the data. Hypothesis testing is a statistical technique that is used to help make inferences about a population parameter. Hypothesis testing allows you to test whether a claim about a parameter is accurate or not.

Your boss makes the claim that the average body fat in men attending Silver's Gym is 20%. You believe that the average body fat for men attending Silver's Gym is not 20%. For claims such as this, you can set up a hypothesis test to reach one of two possible conclusions: either a decision cannot be made to disprove the body fat average of 20%, or there is enough evidence to say that the body fat average claim is inaccurate.

To assist in your analysis for Silver's Gym, consider the following steps based on your boss's claim that the mean body fat in men attending Silver's Gym is 20%:

First, construct the null and alternative hypothesis test based on the claim by your boss.

Using an alpha level of 0.05, perform a hypothesis test, and report your findings. Be sure to discuss which test you will be using and the reason for selection. Recall you found the body fat mean and standard deviation in Part I of the task.

Based on your results, interpret the final decision to report to your boss.

Parts I-II: Review and revise your individual project from last week. You must include parts I and II from Individual Project #4 as they will be graded again. Then, add the following responses to your document:

Part III: Regression and Correlation

Based on what you have learned from your research on regression analysis and correlation, answer the following questions about the Body Fat Versus Weight data set:

When performing a regression analysis, it is important to first identify your independent/predictor variable versus your dependent/response variable, or simply put, your x versus y variables. How do you decide which variable is your predictor variable and which is your response variable?

Based on the Body Fat Versus Weight data set, which variable is the predictor variable? Which variable is the response variable? Explain.

Using Excel, construct a scatter plot of your data.

Using the graph and intuition, determine whether there is a positive correlation, a negative correlation, or no correlation. How did you come to this conclusion?

Calculate the correlation coefficient, r, and verify your conclusion with your scatter plot. What does the correlation coefficient determine?

Add a regression line to your scatter plot, and obtain the regression equation.

Does the line appear to be a good fit for the data? Why or why not?

Regression equations help you make predictions. Using your regression equation, discuss what the slope means, and determine the predicted value of weight when body fat equals 0. Interpret the meaning of this result

Part IV: Putting it Together

Your analysis is now complete, and you are ready to report your findings to your boss. In one paragraph, summarize your results by explaining your findings from the statistical measures, hypothesis test, and regression analysis of body fat and weight for the 252 men attending Silver's Gym.

Please submit your assignment.

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#### Solution Summary

The solution gives detailed steps on conducting a case study of normal distributed data including regression analysis and hypothesis testing. All formula and calculations are shown and explained.

Continuous Probability Distributions: Normal Distribution

Fast Service Truck Lines uses the Ford Super Duty F-750 exclusively. Management made a study of the maintenance costs and determined the number of miles traveled during the year followed the normal distribution. The mean of the distribution was 60,000 miles and the standard deviation 2,000 miles.

a. What percent of the Ford Super Duty F-750s logged 65,200 miles or more?

b. What percent of the trucks logged more than 57,060 but less than 58,280 miles?

c. What percent of the Fords traveled 62,000 miles or less during the year?

d. Is it reasonable to conclude that any of the trucks were driven more than 70,000 miles? Explain.