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Chapter 24: Problem 25

Take \(p_{V L B W}\) and \(p_{\text {contrnd }}\) to be the proportions of all VLBW and normal-birth-weight (control) babies who would graduate from high school. The hypotheses to be tested are a. \(H_{0}: p_{V L B W}=p_{\text {control }}\) versus \(H_{a}: p_{V L B W} \neq p_{\text {control. }}\) b. \(H_{0}: p_{V L B W}=p_{\text {control }}\) versus \(H_{a}: p_{V L B W}>p_{\text {control. }}\) c. \(H_{0}: p_{V L B W}=p_{\text {control }}\) versus \(H_{a}: p_{V L B W}p_{\text {control }}\) versus \(H_{a}: p_{V L B W}=p_{\text {controi }}\)

Short Answer

Expert verified
Parts (a), (b), and (c) present valid hypotheses, while part (d) is incorrectly set up.

Step by step solution

01

Define Proportions

We begin by understanding that \(p_{VLBW}\) is the proportion of very low birth weight (VLBW) babies who graduate high school, and \(p_{control}\) is the proportion of normal-birth-weight (control) babies who graduate from high school.
02

Hypothesis for Part (a)

The null hypothesis is \(H_0: p_{VLBW} = p_{control}\), indicating that there is no difference in graduation proportions between VLBW and control babies. The alternative hypothesis is \(H_a: p_{VLBW} eq p_{control}\), suggesting that there is a difference in graduation proportions.
03

Hypothesis for Part (b)

For part (b), we have the null hypothesis \(H_0: p_{VLBW} = p_{control}\) indicating equality of proportions. The alternative hypothesis is \(H_a: p_{VLBW} > p_{control}\), suggesting that VLBW babies have a higher graduation proportion than control babies.
04

Hypothesis for Part (c)

In part (c), the null hypothesis is \(H_0: p_{VLBW} = p_{control}\). The alternative hypothesis \(H_a: p_{VLBW} < p_{control}\) implies that the graduation proportion for VLBW babies is less than that of the control group.
05

Hypothesis for Part (d) - Incorrect Setup

In part (d), the statement \(H_0: p_{VLBW} > p_{control}\) and \(H_a: p_{VLBW} = p_{control}\) is a non-standard hypothesis testing setup. Typically, the null hypothesis should reflect no difference or equality, thus making this setup incorrect.

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Key Concepts

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

Proportion Comparison
Proportion comparison is a fundamental concept in statistics. In this context, it helps us understand the differences or similarities in two different groups. For our exercise, the groups are VLBW (very low birth weight) babies and normal-birth-weight (control) babies. Each group has a proportion that represents how many in that group graduate high school. These are denoted as \(p_{VLBW}\) and \(p_{control}\).
  • Proportion: A type of ratio that represents a part of a whole. It is expressed as a fraction of the total.
  • Comparison: To determine if two proportions are equal or different and by how much.
By comparing proportions, we make informed decisions about whether interventions or variations in circumstances lead to different outcomes in populations.
Null Hypothesis
The null hypothesis (denoted as \(H_0\)) is a key element in hypothesis testing. It is the default position that there is no effect or no difference. In our context, the null hypothesis is set as \(p_{VLBW} = p_{control}\). This means we start with the assumption that there is no difference in graduation rates between VLBW and control babies.
  • Significance: Plays a crucial role in the validity of a statistical test.
  • Purpose: To provide a non-biased starting point for statistical instructions.
The null hypothesis goes through testing to determine if there is enough evidence to reject it in favor of an alternative hypothesis, which suggests some effect or difference.
Alternative Hypothesis
The alternative hypothesis (\(H_a\)) directly contrasts the null hypothesis. It showcases the specific difference or effect you鈥檙e expecting to prove. Here, it is where we imply differences in the graduation rates between VLBW and control babies.- For instance, one form of the alternative hypothesis is \(H_a: p_{VLBW} eq p_{control}\), indicating any difference in graduation rates.- Another form, \(H_a: p_{VLBW} > p_{control}\), suggests VLBW babies have a higher graduation rate.- Conversely, \(H_a: p_{VLBW} < p_{control}\) suggests they have a lower rate.
  • Nature: Represents the hypothesis you鈥檙e testing against the null.
  • Expectation: Indicates what you suspect is true or what you want to prove.
Understanding the appropriate alternative hypothesis helps to clarify the research question and guides the analysis.
Graduation Rates
Graduation rates are a specific measure reflecting the proportion of individuals who complete high school. In hypothesis testing, these rates are converted into proportions, such as \(p_{VLBW}\) and \(p_{control}\), to facilitate statistical analysis.
  • Indicator: Reflects educational outcomes within a population.
  • Analysis: Used as a benchmark to compare different populations.
By focusing on graduation rates, stakeholders can assess the impact of varying health conditions or interventions on educational attainment within differing groups.
Statistical Significance
Statistical significance helps us understand if the findings from our hypothesis testing are compelling enough to reject the null hypothesis. It's a measure indicating the likelihood that the results, such as differences in graduation rates, are due to some factor other than random chance.
  • Threshold: Usually set at a standard level like \(\alpha = 0.05\).
  • Interpretation: If results are statistically significant, it suggests a real effect or difference.
Determining statistical significance is vital as it provides confidence in the conclusions drawn from data analysis, thus helping validate or refute assumptions based on empirical evidence.

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Most popular questions from this chapter

Starting to Talk. At what age do infants speak their first word of English? Here are data on 20 children (ages in months): 18 | 1STWORD (In fact, the sample contained one more child, who began to speak at 42 months. Child development experts consider this abnormally late, so the investigators dropped the outlier to get a sample of "typical" children. We are willing to treat these data as an SRS.) Is there good evidence that the mean age at first word among all typical children is greater than one year?

of the 126 women in the VLBW group, 38 said they had used illegal drugs; 54 of the 124 control group women had done so. The IQ scores for the VLBW women who had used illegal drugs had mean \(86.2\) (standard deviation 13.4), and the normal-birth-weight controls who had used illegal drugs had mean IQ \(89.8\) (standard deviation 14.0). Is there a statistically significant difference between the two groups in mean IQ? The P-value for this test is a. less than \(0.01\). b. between \(0.01\) and \(0.05\). c. between \(0.05\) and \(0.10\). d. greater than \(0.10\).

Case for the Supreme Court. In 1986, a Texas jury found a Black man guilty of murder. The prosecutors had used "peremptory challenges" to remove 10 of the 11 Blacks and four of the 31 Whites in the pool from which the jury was chosen. \(\frac{24}{}\) The law says that there must be a plausible reason (that is, a reason other than race) for different treatment of Blacks and Whites in the jury pool. When the case reached the Supreme Court 17 years later, the Court said that "happenstance is unlikely to produce this disparity." The inferential methods we have studied cannot safely be used to support the Court's finding that chance is unlikely to produce so large a Black-White difference. Why not?

Preventing Drowning. Drowning in bathtubs is a major cause of death in children less than five years old. A random sample of parents was asked many questions related to bathtub safety. Overall, \(85 \%\) of the sample said they used baby bathtubs for infants. Estimate the percentage of all parents of young children who use baby bathtubs.

Gastric Bypass Surgery. How effective is gastric bypass surgery in maintaining weight loss in extremely obese people? Researchers found that 427 of 771 subjects who had received gastric bypass surgery regained at least \(25 \%\) of their post-surgery weight loss five years after surgery. 4 a. Are the conditions for the use of the large-sample confidence interval met? Explain. b. Give a \(90 \%\) confidence interval for the proportion of those receiving gastric bypass surgery who regained at least \(25 \%\) of their post-surgery weight loss five years after surgery. c. Interpret your interval in the context of the problem.
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