Question

1. mp modeling real life the table shows the heights of the volleyball players on two teams. compare the mean heights of the two teams. do outliers affect either mean? explain.

player heights (inches)
dolphins: 59, 65, 53, 56, 58, 61, 64, 68, 51, 56, 54, 57
tigers: 63, 68, 66, 58, 54, 55, 61, 62, 53, 70, 64, 64

Explanation:

Step1: Calculate the sum of Dolphins' heights

First, we list the heights of Dolphins: 59, 65, 53, 56, 58, 61, 64, 68, 51, 56, 54, 57.
Sum these values: \(59 + 65 + 53 + 56 + 58 + 61 + 64 + 68 + 51 + 56 + 54 + 57\)
Let's calculate step by step:
\(59+65 = 124\); \(124+53 = 177\); \(177+56 = 233\); \(233+58 = 291\); \(291+61 = 352\); \(352+64 = 416\); \(416+68 = 484\); \(484+51 = 535\); \(535+56 = 591\); \(591+54 = 645\); \(645+57 = 702\)
So the sum of Dolphins' heights is \(702\) inches, and there are \(12\) players.

Step2: Calculate the mean height of Dolphins

The formula for the mean \(\bar{x}=\frac{\sum x}{n}\), where \(\sum x\) is the sum of the data and \(n\) is the number of data points.
For Dolphins: \(\bar{x}_{Dolphins}=\frac{702}{12}=58.5\) inches.

Step3: Calculate the sum of Tigers' heights

List the heights of Tigers: 63, 68, 66, 58, 54, 55, 61, 62, 53, 70, 64, 64.
Sum these values: \(63 + 68 + 66 + 58 + 54 + 55 + 61 + 62 + 53 + 70 + 64 + 64\)
Calculate step by step:
\(63+68 = 131\); \(131+66 = 197\); \(197+58 = 255\); \(255+54 = 309\); \(309+55 = 364\); \(364+61 = 425\); \(425+62 = 487\); \(487+53 = 540\); \(540+70 = 610\); \(610+64 = 674\); \(674+64 = 738\)
So the sum of Tigers' heights is \(738\) inches, and there are \(12\) players.

Step4: Calculate the mean height of Tigers

Using the mean formula: \(\bar{x}_{Tigers}=\frac{738}{12}=61.5\) inches.

Step5: Analyze the effect of outliers

First, identify outliers. For Dolphins, the data points are mostly around 50 - 68. The values 51, 53, 54, 56, 57, 58, 59, 61, 64, 65, 68. Wait, 51 is a bit low, but let's check the range. For Tigers, the data points: 53, 54, 55, 58, 61, 62, 63, 64, 64, 66, 68, 70. 70 is a bit high.
Now, recalculate the mean of Dolphins without 51 (if we consider 51 as an outlier). Sum without 51: \(702 - 51=651\), number of data points \(11\). New mean: \(\frac{651}{11}\approx59.18\), which is close to the original mean of \(58.5\).
For Tigers, recalculate the mean without 70. Sum without 70: \(738 - 70 = 668\), number of data points \(11\). New mean: \(\frac{668}{11}\approx60.73\), which is close to the original mean of \(61.5\). So outliers have a small effect on the means because the number of data points is relatively large (12), and the outliers are not extremely far from the other data points or the effect is diluted by the number of data points. Also, comparing the means: The mean height of Tigers (\(61.5\) inches) is greater than the mean height of Dolphins (\(58.5\) inches).

Answer:

The mean height of Dolphins is \(58.5\) inches, and the mean height of Tigers is \(61.5\) inches. The mean height of Tigers is greater than that of Dolphins. Outliers have a small effect on the means because the number of data points is relatively large, and the outliers' influence is diluted (or the outliers are not extremely far from the main data cluster).