/*! 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 3 A college administrator question... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 3

A college administrator questions the first 50 students he meets on campus the day after final exams are over. He asks them whether they had positive, neutral, or negative overall feelings about the term that had just ended. Suggest some reasons it may be risky to act as if the first 50 students at this particular time are an SRS of all students at this college.

Short Answer

Expert verified
The sample is not random due to timing, location, and size biases, affecting its representativeness.

Step by step solution

01

Understanding the Sample

The administrator questions the first 50 students he meets. This sample is not random because it only includes students the administrator happens to encounter, potentially introducing bias.
02

Timing Consideration

The questioning occurs the day after final exams, which may influence students' feelings, especially if they are relieved or stressed due to recently finishing exams.
03

Location Bias

The location where the administrator meets students could lead to skewed results if it isn't diverse in terms of student representation (e.g., if conducted near an academic building, it might overrepresent certain groups of students).
04

Sample Size Limitation

The sample size of 50 students may not be large enough to accurately represent the diverse feelings and backgrounds of all students at the college.

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.

Random Sampling
Random sampling is an essential concept in survey research. It means selecting participants in a way that every individual in the population has an equal chance of being chosen. In practice, this helps ensure that the sample represents the whole group without bias. When done properly, random sampling can prevent biased results, which occur when only certain types of individuals are included more than others.

Failure to use random sampling was evident in the administrator's approach. By questioning only the first students he met, he might have unintentionally introduced sampling bias. These students may share similar characteristics鈥攍ike early risers or those in specific areas of campus鈥攚hich may not represent the wider student body.
  • Random sampling helps mirror the population's diversity.
  • It avoids the influence of personal biases.
  • Ensures credibility and validity of the survey findings.
Sample Size
The sample size, which refers to the number of participants in a study, greatly impacts the accuracy of survey results. A larger sample size often provides more reliable data, reducing the margin of error. It helps capture a broader spectrum of opinions and minimizes the impact of outliers.

For the college scenario, surveying just 50 students might not adequately capture the varied opinions of the entire student body. Consider the vast number of factors that could affect student opinions, such as differing majors, years in school, and personal experiences.
  • Larger sample sizes increase result precision.
  • Small samples can lead to misleading conclusions.
  • Sample size planning is crucial for data reliability.
Survey Timing
The timing of a survey can significantly influence the responses of participants. People's moods and opinions can fluctuate based on recent events, such as exams or holidays. This is known as a response bias due to timing.

In this case, surveying students the day after finals introduced potential bias. Students might still be feeling stressed or relieved from the exam period, which could color their opinions about the entire term disproportionately. Thus, the timing may affect the survey results, making them not entirely representative.
  • Recent events can bias responses.
  • Timing should ideally reflect regular conditions.
  • Planning neutral times for surveys avoids timing bias.
Location Bias
Location bias arises when the place where a survey is conducted affects who participates and the responses given. People found in specific locations may not represent the broader group accurately. For instance, students near academic buildings might have different sentiments than those in leisure areas.

In the exercise, if the administrator only questions students passing by a library or dormitory, there's a risk of overrepresenting certain student groups. Diverse locations across the campus would provide a more varied and accurate student sample.
  • Different locations attract different groups.
  • Survey diversity improves data representativeness.
  • Strategic location planning is key for unbiased sampling.

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

Does the birth order of a family's children influence their IQ scores? A careful study of 241,310 Norwegian 18- and 19 year-olds found that firstborn children scored \(2.3\) points higher on the average than second children in the same family. This difference was highly significant \((P<0.001)\). A commentator said, "One puzzle highlighted by these latest findings is why certain other within-family studies have failed to show equally consistent results. Some of these previous null findings, which have all been obtained in much smaller samples, may be explained by inadequate statistical power."11 Explain in simple language why tests having low power often fail to give evidence against a null hypothesis even when the hypothesis is really false.

A 2015 Gallup Poll asked a national random sample of 398 adult women to state their current weight. The mean weight in the sample was \(\mathrm{x}^{-} \bar{x}=155\). We will treat these data as an SRS from a Normally distributed population with standard deviation \(\sigma=35\). (a) Give a \(95 \%\) confidence interval for the mean weight of adult women based on these data. (b) Do you trust the interval you computed in part (a) as a 95\% confidence interval for the mean weight of all U.S. adult women? Why or why not?

Finding power by hand. Even though software is used in practice to calculate power, doing the work by hand builds your understanding. Return to the test in Example 18.6. There are \(n=10\) observations from a population with standard deviation \(\sigma=1\) and unknown mean \(\mu\). We will test $$ \begin{aligned} &H_{0}: \mu=0 \\ &H_{a}: \mu>0 \end{aligned} $$ with fixed significance level \(\alpha=0.05\). Find the power against the alternative \(\mu=0.8\) by following these steps. (a) The \(z\) test statistic is $$ \mathrm{z}=\mathrm{x}^{-}-\mu 0 \sigma / \mathrm{n}=\mathrm{x}^{-}-01 / 10=3.162 \mathrm{x}^{-}=\frac{\bar{x}-\mu_{0}}{\sigma / \sqrt{n}}=\frac{\bar{x}-0}{1 / \sqrt{10}}=3.162 \bar{x} $$ (Remember that you won't know the numerical value of \(x^{-} \bar{x}\) until you have data.) What values of \(z\) lead to rejecting \(H_{0}\) at the \(5 \%\) significance level? (b) Starting from your result in part (a), what values of \(x^{-} \bar{x}\) lead to rejecting \(H_{0}\) ? The area above these values is shaded under the top curve in Figure 18.1. (c) The power is the probability that you observe any of these values of \(\mathrm{x}^{-} \bar{x}\) when \(\mu=0.8\). This is the shaded area under the bottom curve in Figure 18.1. What is this probability?

Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by \(\mathrm{pH}\) on a scale of 0 to 14 . Distilled water has \(\mathrm{pH} 7.0\), and lower \(\mathrm{pH}\) values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a \(\mathrm{pH}\) below 5.0. Suppose that \(\mathrm{pH}\) measurements of rainfall on different days in a Canadian forest follow a Normal distribution with standard deviation \(\sigma=0.6\). A sample of \(n\) days finds that the mean \(\mathrm{pH}\) is \(\mathrm{x}^{-} \bar{x}=4\).8. Is this good evidence that the mean \(\mathrm{pH} \mu\) for all rainy days is less than 5.0? The answer depends on the size of the sample. Either by hand or using the P-Value of a Test of Significance applet, carry out four tests of $$ \begin{aligned} &H_{0}: \mu=5.0 \\ &H_{a}: \mu<5.0 \end{aligned} $$ Use \(\sigma=0.6\) and \(x^{-} \bar{x}=4.8\) in all four tests. But use four different sample sizes: \(n=\) \(9, n=16, n=36\), and \(n=64\). (a) What are the \(P\)-values for the four tests? The P-value of the same result \(x^{-} \bar{x}\) \(=4.8\) gets smaller (more significant) as the sample size increases. (b) For each test, sketch the Normal curve for the sampling distribution of \(x^{-} \bar{x}\) when \(H_{0}\) is true. This curve has mean \(5.0\) and standard deviation \(0.6 / \mathrm{n} \sqrt{n}\). Mark the observed \(x^{-} \bar{x}=4.8\) on each curve. (If you use the applet, you can just copy the curves displayed by the applet.) The same result \(x-\bar{x}=4.8\) gets more extreme on the sampling distribution as the sample size increases.

Many sample surveys use well-designed random samples, but half or more of the original sample can't be contacted or refuse to take part. Any errors due to this nonresponse (a) have no effect on the accuracy of confidence intervals. (b) are included in the announced margin of error. (c) are in addition to the random variation accounted for by the announced margin of error.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

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.