Data set captured over a 35 month period between October ...



Purpose: To determine if the shoe sizes of the population of men are correlated to their heights (This information will help the shoe manufacturers to plan their spending on materials, and enable them to deliver products that will meet the customer demands and expectations).

(A) A random sample of 40 men students was selected from across different localities of Chicago. Their heights (in m) was recorded. They were asked what their shoe size (American shoe size standards for men) was and even this data was recorded. The data was tabulated as shown below.

| |Height In Meters |American Shoe |

| | |size for Men |

|1 |1.8 |10 |

|2 |1.8 |8.5 |

|3 |1.82 |13 |

|4 |1.9 |10 |

|5 |1.7 |8.5 |

|6 |1.82 |11 |

|7 |1.75 |8.5 |

|8 |1.78 |10 |

|9 |1.71 |8.5 |

|10 |1.81 |8.5 |

|11 |1.82 |12 |

|12 |1.71 |8 |

|13 |1.81 |9 |

|14 |1.82 |10 |

|15 |1.71 |10 |

|16 |1.74 |11 |

|17 |1.8 |8.5 |

|18 |1.79 |10 |

|19 |1.78 |9 |

|20 |1.76 |8.5 |

|21 |1.7 |7.5 |

|22 |1.65 |8 |

|23 |1.71 |7.5 |

|24 |1.77 |13 |

|25 |1.89 |12 |

|26 |1.73 |8 |

|27 |1.78 |10 |

|28 |1.8 |10 |

|29 |1.76 |7.5 |

|30 |1.72 |8.5 |

|31 |1.7 |8.5 |

|32 |1.76 |8.5 |

|33 |1.73 |8.5 |

|34 |1.73 |8.5 |

|35 |1.74 |8.5 |

|36 |1.8 |8.5 |

|37 |1.72 |6 |

|38 |1.8 |8.5 |

|39 |1.73 |8 |

|40 |1.91 |10 |

(B) Descriptive statistics such as mean, median, quartiles, and box plots were obtained for both height and shoe size to get a feel of the data. Some insight into the population statistics can be obtained using the sample statistics and a suitable confidence level. For example, from the sample statistics of height, we notice that the mean height is 1.77 m and the mean shoe size is 9.2 (The nearest higher standard shoe size is 9.5). The population of men can be expected to have heights in the range [1.75 m, 1.79 m] and a shoe size in the range [8.7 and 9.7]. Even though this information may not be directly relevant to the current study, it is never-the-less a useful tool in the overall analysis.

A box plot of both the variables shows the quartiles, the median and outliers. The p- values for a goodness of fit test are less than ( = 0.05 for both the variables implying that heights as well as shoe sizes have normal distributions.

(C) A scatter plot of heights vs shoe sizes is drawn to get an initial idea about any possible association between the two variables. The points appear to lie scattered about a line but some points are scattered far away from the line. This may indicate only a moderate association between height and shoe sizes.

(D) A least squares regression line is then fit to the points in the scatter plot (Height in m is the independent variable, x and Shoe size is the dependent variable, y). Its equation is y = 14.56x - 16.56, that is Shoe Size = 14.56 (Height in m) - 16.56. The correlation coefficient is r = 0.552. This confirms our initial hunch of a moderate correlation between heights and shoe sizes. r^2 = 0.304 means only about 30.4% of the variation in shoe size is explained by variation in height. Shoe size may not after all depend entirely on the height.

(All the results and plots are shown in the excel file with the same name.)

(E) The correlation coefficient r = 0.552 was then compared to the critical value at a 0.05 significance level obtained from the Table of Critical Values for the Pearson Correlation Coefficient. The critical value with n = 40 (number of pairs in the data set - and therefore n - 2 = 40 - 2 = 38 degrees of freedom) is 0.325. (We use 35 degrees of freedom since 38 is not available in the table.) r = 0.325 is the lowest value of r that is required for confirming that heights and shoe sizes are correlated. Since our r- value (0.552) > 0.325, it appears that there exists a correlation between heights and shoe sizes.

(F) The correlation hypothesis for the population was then formulated and tested as shown below. The null hypothesis will be that shoe size and height are not correlated whereas alternate hypothesis will be that they are correlated.

H0: Heights and shoe sizes are not correlated, that is ( = 0

vs

Ha: Heights and shoe sizes are correlated, that is ( ≠ 0

t- test at a level of significance of α= 0.05

Degrees of freedom is.= 40 - 2 = 38

Two-tailed critical t-scores are -2.0244 and 2.0244. The critical region is shown shaded in the diagram below:

[pic]

Decision Rule: Reject H0 if the test t- score < -2.0244 or > 2.0244

The calculated t- score = r * ({(n - 2)/(1 - r^2)} = 0.552 * ({(40 - 2)/(1 - 0.552^2)} = 4.0808

Since 4.0808 > 2.0244, we reject H0 and accept Ha

Conclusion: There is enough evidence at the 5% level of significance to conclude that there is a correlation in the population of men between heights and shoe sizes. But as said earlier, since the r- value (0.552) is low, it appears that there might be other factors that affect the shoe size. One factor that might explain the not so high association is that the shoes come in standard sizes (this makes the data ordinal) and a person has to reconcile with the nearest size available.

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