# Summary statistics

2. (no explanation required) suppose we have a set of sample data. For each quantity listed below, could the quantity be negative? (For each quantity, answer Yes or No.)

Quantity Possibly Negative? (Yes or No)

Arithmetic Mean

Standard Deviation

Frequency

Median

Mode

Range

First Quartile

Interquartile Range

3. The stopping distance on a wet surface was determined for 17 cars each traveling at 30 miles an hour. The stopping distances (in feet) are:

63 69 85 65 90 66 71 75 84

104 87 72 85 85 71 73 91

(a) Find the median. Show work/explanation.

(b) Find the mode.

(c) Find the first quartile. Show work/explanation.

(d) Find the third quartile. Show work/explanation.

(e) Find the range.

(f) Find the interquartile range.

(g) Find the 90th percentile. Show work/explanation.

4. For each of six days, the outdoor temperature was recorded at noon, and the following temperatures were recorded (and listed in ascending order):

8 39 41 45 49 52

(a) Find the median temperature.

(b) Find the mean temperature.

(c) What is the shape of the distribution of temperatures? (skewed left, symmetric, or skewed right?)

(d) Compute the sample variance and sample standard deviation, completing the table below.

8

39

41

45

49

52

Sample variance: ___________

Sample standard deviation: ___________

(e) Compute the coefficient of variation (CV).

(f) Suppose we discover that there has been an error in recording the temperatures. Suppose that instead of 8, it should have been 38, so that the data are actually

38 39 41 45 49 52

If we redo our calculations, indicate what we should expect, by answering the following:

(i) Will the newly calculated mean be lower, the same, or higher than before? _________

(ii) Will the newly calculated median be lower, the same, or higher than before? ________

(iii) Will the newly calculated sample standard deviation be lower, the same, or higher than before? _______

5. The (fictional) university Central University has two professional graduate programs, the Business School and the Law School. The following tables show data about 700 male applicants and about 500 female applicants.

Table M: MALE Applicants

Admitted Not admitted Total

Business School 480 120

Law School 10 90

Total

Table F: FEMALE Applicants

Admitted Not admitted Total

Business School 180 20

Law School 100 200

Total

(a) Fill in the totals in Table M and in Table F.

(b) For Table M, construct a table of row percentages. For Table F, construct a table of row percentages.

MALE Applicants

Admitted Not admitted Total

Business School

Law School

Total

FEMALE Applicants

Admitted Not admitted Total

Business School

Law School

Total

(b) Looking at the Business School, who is admitted at a higher rate, men or women?

(c) Looking at the Law School, who is admitted at a higher rate, men or women?

(d) Looking at the last row, the totals for both business and law schools combined, who is admitted at a higher rate, men or women?

(e) Why is the answer for (d) so different than for (b) and (c)? Can you see what is causing this effect? Discuss briefly.

HINT: Try constructing tables for the total percentages and see if that provides additional insight.

MALE Applicants

Admitted Not admitted Total

Business School

Law School

Total

FEMALE Applicants

Admitted Not admitted Total

Business School

Law School

Total

6. Dinner check amounts at a particular restaurant have the frequency distribution shown below.

Dinner Frequency

Check

$25 - under $35 3

$35 - under $45 9

$45 - under $55 13

$55 - under $65 20

$65 - under $75 15

Total 60

Compute the estimated mean dinner check amount for these grouped data. Show work.

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See attached file for complete description of the questions

https://brainmass.com/statistics/quantative-analysis-of-data/summary-statistics-49782

#### Solution Summary

Answers questions on summary statistics-mean, median, mode, first quartile, third quartile, range, interquartile range, 90th percentile, shape of the distribution, skewness, sample variance and sample standard deviation, coefficient of variation, grouped data

Summary statistics, t test, Regression and Correlation: Silver's Gym and Body Fat vs. Weight

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.

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. 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 body fat (y) when weight (x) equals 0. Interpret the meaning of this equation.

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.

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