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Chapter 8: Problem 71

A state wildlife service wants to estimate the mean number of days that each licensed hunter actually hunts during a given season, with a bound on the error of estimation equal to 2 hunting days. If data collected in earlier surveys have shown \(\sigma\) to be approximately equal to 10 , how many hunters must be included in the survey?

Short Answer

Expert verified
97 hunters should be included in the survey.

Step by step solution

01

Identify the given values

We are given the standard deviation \(\sigma = 10\), and the margin of error \(E = 2\). Since no confidence level is mentioned, we will assume a common confidence level, such as 95%, which corresponds to a critical value \(z\). For a 95% confidence level, the critical value \(z\) is approximately 1.96.
02

Set up the formula for sample size

The formula to determine the sample size \(n\) for estimating the mean with a specified margin of error is given by: \[ n = \left( \frac{z \cdot \sigma}{E} \right)^2 \]. We will substitute the known values for \(z\), \(\sigma\), and \(E\) into this formula.
03

Substitute the values into the formula

Substitute \(z = 1.96\), \(\sigma = 10\), and \(E = 2\) into the formula: \[ n = \left( \frac{1.96 \cdot 10}{2} \right)^2 \].
04

Calculate the sample size

First, calculate the expression inside the parentheses: \(1.96 \cdot 10 = 19.6\). Then divide by 2: \(\frac{19.6}{2} = 9.8\). Finally, square the result: \(9.8^2 = 96.04\). Since the sample size must be a whole number, we round up to get \(n = 97\).

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Key Concepts

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

Understanding the Margin of Error
In surveys or studies, the margin of error represents the range within which we expect the true population parameter to fall. It's essentially the maximum amount that the sample results are expected to differ from the actual population.
For example, in a wildlife survey estimating the mean number of hunting days, a margin of error of 2 days means our estimate could be 2 days higher or lower in comparison to the actual average.
  • The margin of error influences the required sample size: a smaller margin of error often requires a larger sample size.
  • It helps reflect the precision of the survey results; a smaller margin indicates higher precision.
Understanding the margin of error is crucial because it determines how reliable the results of a survey are and helps interpret whether the findings are statistically significant.
The Role of Confidence Level
The confidence level in a study indicates the degree of certainty that the true parameter is within the margin of error. It鈥檚 usually expressed as a percentage, such as 95%.
A 95% confidence level means if the same survey were conducted 100 times, the interval estimate (counted with the margin of error) would contain the true population parameter 95 times.
  • Common confidence levels are 90%, 95%, and 99%.
  • Higher confidence levels widen the margin of error, thus requiring a larger sample size for greater certainty.
In our wildlife survey example, assuming a 95% confidence level implies that we have a high level of assurance in our estimate for the hunters' mean number of hunting days, considering the margin of error provided.
Exploring Standard Deviation
Standard deviation (\(\sigma\)) measures the amount of variation or dispersion in a set of values. In our wildlife survey example, the given standard deviation is 10 days, indicating hunters' hunting days vary by this amount on average.
A higher standard deviation suggests more variation in the data, which means the data points are more spread out from the mean. A lower standard deviation indicates that the data points tend to be closer to the mean.
  • It plays a critical role in determining the sample size; more variation in data usually demands a larger sample to accurately estimate the population mean.
  • Standard deviation aids in understanding the consistency of a dataset.
Grasping the concept of standard deviation enriches our understanding of how much individual hunters' behavior might differ from the average days hunted.
Conducting a Wildlife Survey
Wildlife surveys are systematic investigations where biologists and researchers gather data about wildlife populations. Surveys like the one described help in understanding hunter behaviors and influence wildlife management strategies.
To conduct a successful wildlife survey:
  • Define clear objectives: Determine exactly what information you want to collect, such as hunting patterns or population estimates.
  • Choose an appropriate sample size: The sample must be large enough to represent the population adequately, influenced greatly by factors like margin of error and standard deviation.
  • Use random sampling methods: This helps ensure that the survey results are unbiased and representative of the entire population.
Conducting these surveys accurately results in better insights into wildlife trends and supports sustainable wildlife conservation efforts.

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Most popular questions from this chapter

What is the normal body temperature for healthy humans? A random sample of 130 healthy human body temperatures provided by Allen Shoemaker \(^{\star}\) yielded 98.25 degrees and standard deviation 0.73 degrees. a. Give a \(99 \%\) confidence interval for the average body temperature of healthy people. b. Does the confidence interval obtained in part (a) contain the value 98.6 degrees, the accepted average temperature cited by physicians and others? What conclusions can you draw?

An investigator is interested in the possibility of merging the capabilities of television and the Internet. A random sample of \(n=50\) Internet users yielded that the mean amount of time spent watching television per week was 11.5 hours and that the standard deviation was 3.5 hours. Estimate the population mean time that Internet users spend watching television and place a bound on the error of estimation.

Suppose that \(Y_{1}, Y_{2}, Y_{3}, Y_{4}\) denote a random sample of size 4 from a population with an exponential distribution whose density is given by $$f(y)\left\\{\begin{array}{ll} (1 / \theta) e^{-y / \theta}, & y>0 \\ 0, & \text { elsewhere } \end{array}\right.$$ a. Let \(X=\sqrt{Y_{1} Y_{2}}\). Find a multiple of \(X\) that is an unbiased estimator for \(\theta\). [Hint: Use your knowledge of the gamma distribution and the fact that \(\Gamma(1 / 2)=\sqrt{\pi}\) to find \(E(\sqrt{Y_{1}})\). Recall that the variables \(Y_{i}\) are independent.] b. Let \(W=\sqrt{Y_{1} Y_{2} Y_{3} Y_{4}}\). Find a multiple of \(W\) that is an unbiased estimator for \(\theta^{2}\). [Recall the hint for part (a).]

Refer to Exercise 8.34. In polycrystalline aluminum, the number of grain nucleation sites per unit volume is modeled as having a Poisson distribution with mean \(\lambda\). Fifty unit-volume test specimens subjected to annealing under regime A produced an average of 20 sites per unit volume. Fifty independently selected unit-volume test specimens subjected to annealing regime \(\mathrm{B}\) produced an average of 23 sites per unit volume. a. Estimate the mean number \(\lambda_{\mathrm{A}}\) of nucleation sites for regime \(\mathrm{A}\) and place a 2 -standard-error bound on the error of estimation. b. Estimate the difference in the mean numbers of nucleation sites \(\lambda_{A}-\lambda_{B}\) for regimes \(A\) and B. Place a 2-standard-error bound on the error of estimation.Would you say that regime B tends to produce a larger mean number of nucleation sites? Why?

The Mars twin rovers, Spirit and Opportunity, which roamed the surface of Mars in the winter of 2004, found evidence that there was once water on Mars, raising the possibility that there was once life on the plant. Do you think that the United States should pursue a program to send humans to Mars? An opinion poll \(^{\star}\) indicated that \(49 \%\) of the 1093 adults surveyed think that we should pursue such a program. a. Estimate the proportion of all Americans who think that the United States should pursue a program to send humans to Mars. Find a bound on the error of estimation. b. The poll actually asked several questions. If we wanted to report an error of estimation that would be valid for all of the questions on the poll, what value should we use? [Hint: What is the maximum possible value for \(p \times q ?]\)
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