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Chapter 3: Problem 23

Florida Lakes Florida has over 7700 lakes. \(^{12}\) We wish to estimate the correlation between the pH levels of all Florida lakes and the mercury levels of fish in the lakes. We see in Data 2.4 on page 71 that the correlation between these two variables for a sample of \(n=53\) of the lakes is -0.575 . (a) Give notation for the quantity we are estimating, notation for the quantity we use to make the estimate, and the value of the best estimate. (b) Why is an estimate necessary here? What would we have to do to calculate the exact value of the quantity we are estimating?

Short Answer

Expert verified
a) We're estimating the population correlation (\(\rho\)), using the sample correlation (r) as our tool. The best estimate is -0.575. b) An estimate is necessary because we only tested 53 out of over 7700 lakes. To calculate the exact value, we would need to measure the pH levels and mercury levels in fish from every lake in Florida.

Step by step solution

01

Identifying the symbols

The quantity we wish to estimate is the population correlation - denoted as rho (\(\rho\)). The quantity we are using to estimate this is the sample correlation, denoted using r.
02

Determining the value of the best estimate

The best estimate for the population correlation is the sample correlation we have, which is -0.575.
03

Understanding the need for an estimate

An estimate is necessary here because we only have the pH levels and mercury levels for 53 lakes out of more than 7700 lakes in Florida. The sample data allows us to make an educated estimate about the population on the whole.
04

Computing the exact value

In order to calculate the exact value of the quantity we are estimating, we would need to gather pH levels and mercury levels for all 7700 lakes, which might not be practically feasible, hence the need for an estimate based on the sample data.

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Most popular questions from this chapter

SKILL BUILDER 1 In Exercises 3.41 to \(3.44,\) data from a sample is being used to estimate something about a population. In each case: (a) Give notation for the quantity that is being estimated. (b) Give notation for the quantity that gives the best estimate. Random samples of organic eggs and eggs that are not organic are used to estimate the difference in mean protein level between the two types of eggs.

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Exercises 3.96 to 3.101 use data from a study designed to examine the effect of doing synchronized movements (such as marching in step or doing synchronized dance steps) and the effect of exertion on many different variables, such as pain tolerance and attitudes toward others. In the study, 264 high school students in Brazil were randomly assigned to one of four groups reflecting whether or not movements were synchronized (Synch= yes or no) and level of activity (Exertion= high or low). \(^{49}\) Participants rated how close they felt to others in their group both before (CloseBefore) and after (CloseAfter) the activity, using a 7-point scale (1=least close to \(7=\) most close ). Participants also had their pain tolerance measured using pressure from a blood pressure cuff, by indicating when the pressure became too uncomfortable (up to a maximum pressure of \(300 \mathrm{mmHg}\) ). Higher numbers for this Pain Tolerance measure indicate higher pain tolerance. The full dataset is available in SynchronizedMovement. For each of the following problems: (a) Give notation for the quantity we are estimating, and define any relevant parameters. (b) Use StatKey or other technology to find the value of the sample statistic. Give the correct notation with your answer. (c) Use StatKey or other technology to find the standard error for the estimate. (d) Use the standard error to give a \(95 \%\) confidence interval for the quantity we are estimating. (e) Interpret the confidence interval in context. How Close Do You Feel to Others? Use the closeness ratings before the activity (CloseBefore) to estimate the mean closeness rating one person would assign to others in a group.
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