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Chapter 18: Problem 30

A market researcher chooses at random from women entering a large upscale department store. One outcome of the study is a \(95 \%\) confidence interval for the mean of "the highest price you would pay for a handbag." (a) Explain why this confidence interval does not give useful information about the population of all women. (b) Explain why it may give useful information about the population of women who shop at large upscale department stores.

Short Answer

Expert verified
(a) The interval does not reflect all women due to sampling bias. (b) It may be useful for women shopping at upscale stores.

Step by step solution

01

Problem Understanding

We need to explain why a confidence interval from a study with a biased sample may not reflect the general population, but could be more relevant for a more specific subset of the population.
02

Identify the Target Population for Part (a)

The target population for this study ideally includes all women, but the sample is taken from just those entering a large upscale department store.
03

Analyze the Sample Bias for Part (a)

The sample is biased because it only includes women who shop at an upscale department store, which may not represent all women in terms of spending habits. Hence, the confidence interval is not reflective of the entire population of women.
04

Identify the Target Population for Part (b)

For part (b), the target population is women who shop at large upscale department stores. The sample is taken exactly from this population.
05

Consider Sample Relevance for Part (b)

Since the sample aligns with the target population (women who shop at upscale department stores), the confidence interval is more likely to reflect the spending habits of this specific group.

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Key Concepts

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

Sample Bias
Sample bias occurs when the participants selected for a study do not truly represent the entire population of interest. In the given exercise, women entering a large upscale department store are chosen as the sample. This creates a sample bias because these women might have different spending habits compared to all women in general. For instance, their willingness to pay for a handbag could be higher due to socio-economic status or fashion preferences associated with shopping at upscale stores.

To better understand sample bias, consider these key points:
  • Selective Sampling: The sample is drawn from only a specific segment of the population, in this case, upscale department store shoppers.
  • Skewed Results: The results may not generalize to the broader population. Thus, any conclusions about the spending habits of all women based on this sample could be misleading.
Recognizing and minimizing sample bias is crucial for producing valid and reliable research findings.
Population Parameter
A population parameter is a characteristic or measure that describes an aspect of an entire population. In statistical studies, we are often interested in estimating these parameters using data collected from a sample. For instance, the mean highest price women would pay for a handbag is a population parameter in this study.

Understanding population parameters is vital for effective research:
  • Representing the Whole: This parameter aims to reflect all individuals within the study's focus, though true values are often unknowable without surveying the entire population.
  • Estimation Through Sampling: Researchers use sample data to make informed estimates about the population parameter, knowing the sample must be representative for accurate results.
Confidence intervals are tools to help estimate these parameters more precisely, acknowledging some level of uncertainty in our estimates.
Statistical Inference
Statistical inference involves drawing conclusions about a population's characteristics based on a sample's data. It is a key part of research to make predictions or general statements about a broader group, relying on probability and statistical methods.

In the context of the exercise, the role of statistical inference includes:
  • Generalizing Findings: Using a sample to extend conclusions to a larger population, even if direct measurement of the population is not possible.
  • Understanding Confidence Intervals: A confidence interval gives a range in which we expect the population parameter to fall, providing a measure of reliability in our predictions.
Effective statistical inference hinges on high-quality samples. If a sample is biased, as in the exercise scenario, it undermines the validity of any inferences made. Thus, ensuring sample representativeness is essential to reach sound conclusions.

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Most popular questions from this chapter

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.

The power of a test is important in practice because power (a) describes how well the test performs when the null hypothesis is actually true. (b) describes how sensitive the test is to violations of conditions such as Normal population distribution. (c) describes how well the test performs when the null hypothesis is actually not true.

Example \(16.1\) (page 377) described NHANES data on the body mass index (BMI) of 654 young women. The mean BMI in the sample was \(x^{-} \bar{x}=26.8\). We treated these data as an SRS from a Normally distributed population with standard deviation \(\sigma=7.5\). (a) Suppose that we had an SRS of just 100 young women. What would be the margin of error for \(95 \%\) confidence? (b) Find the margins of error for \(95 \%\) confidence based on SRSs of 400 young women and 1600 young women. (c) Compare the three margins of error. How does increasing the sample size change the margin of error of a confidence interval when the confidence level and population standard deviation remain the same?

When to use pacemakers. A medical panel prepared guidelines for when cardiac pacemakers should be implanted in patients with heart problems. The panel reviewed a large number of medical studies to judge the strength of the evidence supporting each recommendation. For each recommendation, they ranked the evidence as level A (strongest), B, or \(C\) (weakest). Here, in scrambled order, are the panel's descriptions of the three levels of evidence. 10 Which is A, which B, and which C? Explain your ranking. Evidence was ranked as level ____ when data were derived from \(a\) limited number of trials involving comparatively small numbers of patients or from well-designed data analysis of nonrandomized studies or observational data registries. Evidence was ranked as level ____ if the data were derived from multiple randomized clinical trials involving a large number of individuals. Evidence was ranked as level ____ when consensus of expert opinion was the primary source of recommendation.

The Trial Urban District Assessment (TUDA) measures educational progress within participating large urban districts. TUDA gives a reading test scored from 0 to 500 . A score of 210 is a "basic" reading level for fourth-graders. 6 Suppose scores on the TUDA reading test for fourth-graders in your district follow a Normal distribution with standard deviation \(\sigma=40\). In 2013, the mean score for fourth-graders in your district was 220 . You plan to give the reading test to a random sample of 25 fourth-graders in your district this year to test whether the mean score \(\mu\) for all fourth-graders in your district is still above the basic level. You will therefore test $$ \begin{aligned} &H_{0}: \mu=210 \\ &H_{a}: \mu>210 \end{aligned} $$ If the true mean score is again 220 , on average, students are performing above the basic level. You learn that the power of your test at the \(5 \%\) significance level against the alternative \(\mu=220\) is \(0.346\). (a) Explain in simple language what "power \(=0.346\) " means. (b) Explain why the test you plan will not adequately protect you against deciding that average reading scores in your district are not above basic level.
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