/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 There are two common methods for... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 57

There are two common methods for measuring the concentration of a pollutant in fish tissue. Do the two methods differ, on average? You apply both methods to each fish in a random sample of 18 carp and use (a) the paired \(t\) test for \(\mu_{d}\) (b) the one-sample \(z\) test for \(p\). (c) the two-sample \(t\) test for \(\mu_{1}-\mu_{2}\). (d) the two-sample \(z\) test for \(p_{1}-p_{2}\). (e) none of these.

Short Answer

Expert verified
(a) the paired t test for \( \mu_{d} \).

Step by step solution

01

Identifying the Problem Type

We need to determine if the two methods for measuring concentration differ, on average, using statistical tests. Since the same fish undergoes both methods, this suggests a paired comparison is appropriate.
02

Selecting the Correct Test

Given the problem involves applying two measurement methods to the same subjects (the carp fish), it naturally leads us to consider differences for each pair of measurements. This setup aligns with using a paired \( t \) test, as it evaluates the mean difference between paired observations.
03

Evaluating Test Choices

Let's evaluate the options:- (a) The paired \( t \) test is suitable for paired data as discussed.- (b) The one-sample \( z \) test for \( p \) is applied to proportion data, which doesn't fit our case.- (c) The two-sample \( t \) test assumes two independent samples, whereas we have paired data.- (d) The two-sample \( z \) test for \( p_{1}-p_{2} \) is also used for proportions and independent samples.- (e) None of these suggests none are suitable, which isn't true since option (a) fits.
04

Making the Final Decision

Based on the pairing of data and the type of measurements, the paired \( t \) test is appropriately used to analyze whether the two methods differ on average. Therefore, choice (a) is the correct method.

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 91影视!

Key Concepts

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

Statistical Tests
Statistical tests are important tools in data analysis that help us make decisions about our data. They help us determine if there are differences between groups or if results are due to chance. In the context of our exercise, different statistical tests were considered to evaluate if two pollutant measurement methods in fish tissue differed.
  • Paired t-test: This test is specifically designed for comparing two related samples. Here, each fish is tested using both methods, making it suitable to evaluate the mean difference between paired observations. By examining these differences, we can determine if the average measurement differs between the two methods.
  • One-sample z-test: Typically used for testing the mean of a single sample against a population mean, it focuses on proportions rather than means of paired differences. Thus, this test doesn鈥檛 fit our scenario.
  • Two-sample t-test: Ideal for comparing means from two independent groups. In our case, however, each fish undergoes both tests, making them not independent but rather a pair.
  • Two-sample z-test: Primarily concerned with proportions of two independent samples, and hence not applicable here as we are dealing with paired data.
Understanding these tests helps in selecting the correct analysis method for different data sets.
Measurement Method Comparison
When comparing measurement methods, it's crucial to have a systematic approach to assess whether two methods yield different results. This exercise specifically involves measuring pollutant concentration in fish, and two methods were tested on the same sample of fish.
  • Ensuring Accuracy: Applying both methods to each fish helps in minimizing external variability. This is because any biological differences across different subjects are eliminated since the same subject is used in both methods.
  • Identifying Discrepancies: The use of paired measurements helps in detecting actual discrepancies between method results rather than just random variance from different subjects.
  • Analyzing Differences: Through statistical analyses like paired t-tests, you can confidently assess whether there is a significant difference in results, aiding in selecting the better method for accurate readings.
This approach ensures that the chosen method reliably represents the pollutant levels in fish, facilitating accurate environmental analyses.
Data Analysis in Statistics
Data analysis is a crucial step in extracting meaningful conclusions from raw data. In statistic exercises like the one described, it's all about understanding the story behind the numbers through proper analysis.
  • Data Collection: The first step involves gathering data through consistent experiments, such as applying two measurement methods to the same fish. Accurate data collection is key to useful analysis.
  • Descriptive Analysis: Describing and summarizing the collected data, including assessing the mean or average pollutant level determined by each method, helps in understanding basic patterns and variations.
  • Inferential Analysis: This goes a step further by applying statistical tests (like the paired t-test in our case) to infer whether observed patterns are statistically significant. This helps in making informed decisions based on sample data.
Through structured data analysis, we can validate assumptions, test hypotheses, and ultimately reach sound conclusions that back scientific and practical decision-making.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

State which inference procedure from Chapter \(8,9,\) or 10 you would use. Be specific. For example, you might say, "Two-sample z test for the difference between two proportions." You do not need to carry out any procedures. Which inference method? (a) How do young adults look back on adolescent romance? Investigators interviewed 40 couples in their midtwenties. The female and male partners were interviewed separately. Each was asked about his or her current relationship and also about a romantic relationship that lasted at least two months when they were aged 15 or \(16 .\) One response variable was a measure on a numerical scale of how much the attractiveness of the adolescent partner mattered. You want to find out how much men and women differ on this measure. (b) Are more than \(75 \%\) of Toyota owners generally satisfied with their vehicles? Let's design a study to find out. We'll select a random sample of 400 Toyota owners. Then we'll ask each individual in the sample: "Would you say that you are generally satisfied with your Toyota vehicle?" (c) Are male college students more likely to binge drink than female college students? The Harvard School of Public Health surveys random samples of male and female undergraduates at four-year colleges and universities about whether they have engaged in binge drinking. (d) A bank wants to know which of two incentive plans will most increase the use of its credit cards and by how much. It offers each incentive to a group of current credit card customers, determined at random, and compares the amount charged during the following six months.

Broken crackers We don't like to find broken crackers when we open the package. How can makers reduce breaking? One idea is to microwave the crackers for 30 seconds right after baking them. Breaks start as hairline cracks called "checking." Randomly assign 65 newly baked crackers to the microwave and another 65 to a control group that is not microwaved. After one day, none of the microwave group and 16 of the control group show checking.

Who talks more-men or women? Researchers equipped random samples of 56 male and 56 female students from a large university with a small device that secretly records sound for a random 30 seconds during each 12.5 -minute period over two days. Then they counted the number of words spoken by each subject during each recording period and, from this, estimated how many words per day each subject speaks. The female estimates had a mean of 16,177 words per day with a standard deviation of 7520 words per day. For the male estimates, the mean was 16,569 and the standard deviation was \(9108 .\) Do these data provide convincing evidence of a difference in the average number of words spoken in a day by male and female students at this university?

Young adults living at home A surprising number of young adults (ages 19 to 25 ) still live in their parents' homes. A random sample by the National Institutes of Health included 2253 men and 2629 women in this age group. \({ }^{11}\) The survey found that 986 of the men and 923 of the women lived with their parents. (a) Construct and interpret a \(99 \%\) confidence interval for the difference in the true proportions of men and women aged 19 to 25 who live in their parents' homes. (b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Which of the following is the correct margin of error for a \(99 \%\) confidence interval for the difference in the proportion of male and female college students who worked for pay last summer? (a) \(2.576 \sqrt{\frac{0.851(0.149)}{550}+\frac{0.851(0.149)}{500}}\) (b) \(2.576 \sqrt{\frac{0.851(0.149)}{1050}}\) (c) \(2.576 \sqrt{\frac{0.880(0.120)}{550}+\frac{0.820(0.180)}{500}}\) (d) \(1.960 \sqrt{\frac{0.851(0.149)}{550}+\frac{0.851(0.149)}{500}}\) (e) \(1.960 \sqrt{\frac{0.880(0.120)}{550}+\frac{0.820(0.180)}{500}}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.