91Ó°ÊÓ

For mallard ducks and Canada geese, what percentage of nests are successful (at least one offspring survives)? Studies in Montana, Illinois, Wyoming, Utah, and California gave the following percentages of successful nests (Reference: The Wildlife Society Press, Washington, D.C.). \(x\) : Percentage success for mallard duck nests 56 85 52 13 39 y: Percentage success for Canada goose nests 24 53 60 \(\begin{array}{cc}69 & 18\end{array}\) (a) Use a calculator to verify that \(\Sigma x=245 ; \Sigma x^{2}=14,755 ; \Sigma y=224 ;\) and \(\Sigma y^{2}=12,070\) (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\), the percent of successful mallard nests. (c) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(y\), the percent of successful Canada goose nests. (d) Use the results of parts (b) and (c) to compute the coefficient of variation for successful mallard nests and Canada goose nests. Write a brief explanation of the meaning of these numbers. What do these results say about the nesting success rates for mallards compared to Canada geese? Would you say one group of data is more or less consistent than the other? Explain.

Step by step solution

01

Verification of Summations

Given values: \(x = 56, 85, 52, 13, 39\) and \(y = 24, 53, 60, 69, 18\). We need to verify:1. \(\Sigma x = 56 + 85 + 52 + 13 + 39 = 245\)2. \(\Sigma x^2 = 56^2 + 85^2 + 52^2 + 13^2 + 39^2 = 14755\)3. \(\Sigma y = 24 + 53 + 60 + 69 + 18 = 224\)4. \(\Sigma y^2 = 24^2 + 53^2 + 60^2 + 69^2 + 18^2 = 12070\)These values are already given, confirming the calculation.

02

Calculate Sample Statistics for Mallard Ducks

1. **Sample Mean**: \( \bar{x} = \frac{\Sigma x}{n} = \frac{245}{5} = 49\)2. **Variance**: \( s_x^2 = \frac{\Sigma x^2 - \frac{(\Sigma x)^2}{n}}{n-1} = \frac{14755 - \frac{245^2}{5}}{4} = 798\)3. **Standard Deviation**: \( s_x = \sqrt{s_x^2} = \sqrt{798} \approx 28.25\)

03

Calculate Sample Statistics for Canada Geese

1. **Sample Mean**: \( \bar{y} = \frac{\Sigma y}{n} = \frac{224}{5} = 44.8\)2. **Variance**: \( s_y^2 = \frac{\Sigma y^2 - \frac{(\Sigma y)^2}{n}}{n-1} = \frac{12070 - \frac{224^2}{5}}{4} = 404.7\)3. **Standard Deviation**: \( s_y = \sqrt{s_y^2} = \sqrt{404.7} \approx 20.11\)

04

Calculate Coefficients of Variation

1. **Coefficient of Variation for Mallard Ducks**: \( CV_x = \frac{s_x}{\bar{x}} \times 100 = \frac{28.25}{49} \times 100 \approx 57.65\%\)2. **Coefficient of Variation for Canada Geese**: \( CV_y = \frac{s_y}{\bar{y}} \times 100 = \frac{20.11}{44.8} \times 100 \approx 44.90\%\)

05

Analysis of Consistency

The coefficient of variation (CV) is a relative measure of the dispersion of data. A smaller CV indicates more consistency in the data. In this case, the CV for Canada geese is lower than for mallard ducks (44.90% vs. 57.65%), suggesting that the nesting success rates for Canada geese are more consistent. Thus, the data for Canada geese's nest success rates is more consistent than that for mallard ducks.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean

The sample mean is a measure of the central tendency of a data set. It represents the average value and is calculated by adding all the data points together and dividing by the number of data points. In this exercise, we have two different groups: mallard duck nests and Canada goose nests. For mallard ducks, the data points are the percentages of successful nests: 56, 85, 52, 13, and 39. To find the sample mean (\(\bar{x}\)) of these values, we add them up:

• Sum = 56 + 85 + 52 + 13 + 39 = 245

Now, divide the sum by the number of data points (n = 5):

• Sample Mean (\(\bar{x}\)) = \(\frac{245}{5} = 49\)

Similarly, for Canada geese, we compute the sample mean for 24, 53, 60, 69, and 18:

• Sum = 24 + 53 + 60 + 69 + 18 = 224
• Sample Mean (\(\bar{y}\)) = \(\frac{224}{5} = 44.8\)

This average gives us an idea of what a typical success rate might look like for both species.

Variance

Variance is a measure of how much the values in a data set differ from the mean. It shows the dispersion of the data points.
To find the variance of the mallard duck nesting success rates, we start by calculating the sum of squares of individual deviations from the mean. The formula for the variance (\(s^2\)) is:

• \(s_x^2 = \frac{\Sigma x^2 - \frac{(\Sigma x)^2}{n}}{n-1}\)

For mallard ducks:

• \(\Sigma x^2 = 14755\), and \(\Sigma x = 245\)
• Variance (\(s_x^2\)) = \(\frac{14755 - \frac{245^2}{5}}{4} = 798\)

For Canada geese, using \(\Sigma y = 224\) and \(\Sigma y^2 = 12070\):

• Variance (\(s_y^2\)) = \(\frac{12070 - \frac{224^2}{5}}{4} = 404.7\)

A higher variance means that the data points are spread out more widely around the mean, indicating less consistency in the data.

Standard Deviation

The standard deviation is a widely used measure of variability and is simply the square root of the variance. It is expressed in the same units as the original data, making it easier to interpret compared to variance.
For the sample standard deviation of the mallard duck nest success rates:

• \(s_x = \sqrt{798} \approx 28.25\)

This means the success rates are, on average, about 28.25 percentage points away from the mean success rate of 49%.
For Canada geese:

• \(s_y = \sqrt{404.7} \approx 20.11\)

Here, the data points deviate from their mean of 44.8% by about 20.11 percentage points. Standard deviation helps us understand how tightly knit the data points are to the mean.

Coefficient of Variation

Coefficient of Variation (CV) offers a relative measure of dispersion in the data. It allows for comparison of variation between datasets with different units or scales. CV is calculated as:

• \(CV = \frac{s}{\bar{x}} \times 100\)

For mallard ducks:

• \(CV_x = \frac{28.25}{49} \times 100 \approx 57.65\%\)

For Canada geese:

• \(CV_y = \frac{20.11}{44.8} \times 100 \approx 44.90\%\)

A lower CV indicates more consistency in the data. In this case, the nesting success rates for Canada geese are more consistent (lower CV) than those for mallard ducks. This suggests that there is less variability in the success rates of Canada geese nests compared to mallard ducks.